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There's a lot of talk in the UK these days about how holding a second referendum on Brexit would be "undemocratic" and "go against the spirit of referendums". However repeating a vote doesn't seem all that strange in a democracy. The UK itself holds parliamentary elections every 5 years or even less frequently if the government dissolves parliament. Other countries routinely hold elections every 4 years or even 2 years in the case of US Congress.

So what's the big deal about holding a second Brexit vote? Wouldn't it be akin to choosing a new government after a few years? Why can voters change their mind about who runs the country but cannot change their mind about the outcome of a referendum?

The difference between such one-off referendums and general elections is that everyone knows the general election results are only 'used' for the next term. Indeed, when you vote for a politician in a general election (assuming this takes place in a stable democracy) you know you will get another vote in a certain number of years time.

The difference with the Brexit referendum is that it wasn't clear how it would be implemented. Given the gravity of the matter in the referendum, even those opposed took it seriously, however, it was still a non-binding referendum, it was advisory. The fact that rules weren't / aren't clear to everyone means that people will have different opinions on it. In particular, those who got the result they wanted won't want another referendum and those who think they can win this time might want one.

The big problem with doing such referendums over and over again is that it continues the uncertainty. And even after a second referendum, whatever the outcome, there will be calls for a third, etc. The big problem here is obviously that the rules weren't clear when they had the first referendum. For example, they could have said: we will have a referendum now and then we agree with all parties involved that that result will be it, at least for the next X years. That way, at least everyone knows where they stand and what the result means for them.

Then there's the problem that's specific to Brexit, it's a very complicated issue, much more complicated then installing a couple of hundred new politicians and some staff. As you say, some countries have elections every few years, but in the case of Brexit it has taken that long already to prepare for actually leaving. And that is only the first step, then the UK will want to negotiate their future relationship.

All in all, the problems are twofold: the parties involved hadn't pre-agreed the rules (not even as a 'gentlemen's agreement') and the matter at hand is too complex to change one's position every few years.

When a politician is elected, the voters assume that he or she will be put in office very soon. How many times can you say in a western democracy that a politician was elected, but never actually took office?

The decision by the UK to leave the EU was elected by the voters, but the UK is still in the EU. If the UK holds another referendum and the Remainers win, who's to say that the Leavers can't demand a 3rd referendum? Or 4th? What's the point?

If the subject-matter or circumstances of the "repeat" referendum are sufficiently different, there is no difference from elections.

Yeah, I know the typical argument against referendum repeats is the "democratic deficit" they supposedly have by asking the "same question". (I'll come back to this.) But before we get to that: one could ask the question in reverse: why would a candidate that has failed an election be allowed to run again later for the same office?! After all, the public said no to him. Should he be allowed to run until he gets his way? Of course, even if all candidates are exactly the same on a later ballot, chances are something has changed: their platforms etc.

Now as for the "same question" referendums, the much criticized repeat referendums for EU Treaty changes/adoptions didn't ever have the exact same question, even if it was nominally the same on paper. There were in fact concessions and renegotiations before the "same" question was asked in all cases. The referendum repeat that was preceded by least concessions was the one with the lowest initial turnout: Nice 2001 in Ireland. In that case, the changes were more declarative than legally binding and efforts focused on a better campaign. For the other repeats, the concessions were more substantive and based on the specific objections of the "no" campaigns; after the Danish referendum of 1992, there was a legally binding [under international law] document agreed by all EU heads of state, although its position within the EU legal framework was considered somewhat awkward. The Irish rejection of the Lisbon Treaty in 2008 also resulted in substantive concessions; these took a legal form similar to the solution to the 1992 Danish problem, but it was also boosted by an agreement to include a specific part of it in a future EU treaty, as to clarify its position in EU law as well. (The convenient legal vehicle that was found for the latter was Croatia's accession treaty of 2011.)

Finally, how does this relate to Brexit referendum(s)? The more concrete proposals I know about don't simply ask for a repeat (although I have no doubt the vague notion may have been publicly discussed). Instead the proposal(s) I know about were for a "confirmatory" referendum, meaning the now-concrete deal would be voted on, instead of a general/vague idea. Arguably this is somewhat different than a new referendum following concessions/renegotiation. But in either case, the public has substantially new information that it can use to answer the "same" question, which in the case of a Brexit confirmatory referendum wouldn't even be nominally the same. (Bercow would be pleased.)

And if I'm allowed an imperfect but hopefully still informative analogy: I would compare the 2016 Brexit referendum to an "informal" poll in the EU Council proceedings, which asks each member for their position in principle, at the start of a
discussion, but without binding the member to agree to whatever final legal document is produced. I have a couple of arguments in support of this: unlike the 2011 AV referendum, the law authorizing the 2016 one did not make the referendum result legally binding.
Furthermore, the uncertainty regarding the exact separation conditions (that had yet to be negotiated) give it this "agreement-in-principle" character, similar to the EU Council "informal" polls. (The analogy only goes so far, of course: there is no record kept of the EU Council's informal polls, something that's impossible to do with a nationwide referendum.) The confirmatory referendums proposed (in the amendments I know about) would, in contrast, be legally binding and also about a specific way of exiting the EU.

Finally, I will concede that the topic of repeat referendums is still controversial and does not have a lot of precedents.

At a formal level you're asking about two different standards

Where elections are usually not a source of contention in the UK, the referendum certainly was. Because of a long string of lies and at times dispiriting fear mongering during the campaign (remember the bus that got disowned by Farage the day after Leave won?), and because of campaign finance law violations.

And just to be clear here: even in reasonably well functioning democracies, elections are contested and there's no shortage of campaign finance violations, corruption, and at times outright fraud, or voter suppression, or even legal challenges of the result (remember Gore?). Still, it works relatively well compared to, say, [your pick of a less mature democracy; ideally one with a semi- or full-blown autocrat at the helm].

If you instead compare the Brexit referendum with elections in less well run democracies, where blatant lies and campaign law violations are more commonplace, things become murkier. You'll find that, actually, observers do suggest an honest rerun every so often, when it's not the voters themselves who ask for one. (Whether it gets done is another story entirely of course, which makes it a non-story in western countries unless people get shot at for protesting the vote.)

At a more substantive level there are two sides of this coin

On the one hand side you've May and politicians on both sides of the political spectrum who are arguing that if voters are asked to vote for this again, especially without seeing it put into place before that, then they will lose even more faith in their politicians and their democratic institutions. And I agree with that argument up to a point. You can't put something back to the voters until they give the answer you want (which, fun fact, EU leaders technically did in a sense with the Lisbon treaty). But I'd also stress that in May's specific case, it's also and actually about keeping the Conservative Party intact. And even if we leave the bad faith and the political calculations aside, there's probably something to be said about gaining and retaining voters' trust to begin with.

On the other hand side, those who support a People's Vote argue that hey, those who voted Leave had no idea about what they were signing up for. They were lied to, and promised the moon, etc. They can now actually make up their mind and decide on whether it's a good idea with a concrete deal before them. They actually have an excellent point -- whichever other option(s) voters would want to see on the ballot (No Brexit, or No Deal, or both). And it makes a good deal of sense, when you consider that defending the concrete but imperfect things that you have can't really win a battle in the court of public opinion, against some hypothetical, as of yet defined, and very wishful place some would like to take you.

More to the point, and as I've noted in an earlier comment to your question, the real wtf here is: what's the big deal about repeating a referendum, when the ERG's second in command asked earlier today for an early repeat of May's confidence vote -- on the basis, wait for it, that had they known what it entailed, many would have not voted the same way?