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We see numerous iconic scenes directly lifted from the Tobey Maguire-era Spider-Man films in Spider-Man: Into the Spiderverse, such as:

- the upside-down Kisten Dunst kiss
- stopping a train
- cool Tobey emerging from a New York building and busting a move down the street

etc.

What are in out-of-universe reasons (rights, story-telling, fan-service etc.) for the inclusion of this era and not the eras of Andrew Garfield or Tom Holland?

Christopher Miller: I think the idea is that this Peter Parker is an amalgam of all the Peter Parkers that you have seen in popular culture. So there's elements of the Homecoming Tom Holland Spider-Man, of an Andrew Garfield Spider-Man, of the Tobey Maguire Spider-Man, of Spider-Man from various comics and TV shows. And sort of in this universe the Spider-Man that comes to Miles' world is one that looks similar to but is not exactly the same as the ones that you know. And so, that's why all of those plots are similar, but there's a twist to them.

Phil Lord: "Yeah, the Spider-Man in Miles' universe that he meets early in the movie was meant to be as competent a Spider-Man as possible, and is meant to be living in an alternate universe that we would all consider the mainstream comics universe. So you'll see that like he and M.J. kiss in the rain upside down, but she's upside down, and he's right side up. Just trying to find little ways to say, 'This is a parallel dimension'."

source

As FuzzyBoots pointed out, there is at least one reference to Holland's Spider-Man holding the ferry together.
Also, it seems most likely that Peter's hair is based on the Andrew Garfield version.
It's possible that there were more reference to Maguire's version simply because Maguire was in more Spider-Man films than the other two actors.

According to the film's makers, the aim was to show audiences that this film includes an amalgam of all of the different Spider-Man universes that they're likely to be familiar with.

The specific inclusion of a version of the classic Tobey Maguire "spider-dance" was pushed hard by the co-director because he felt that it gave audiences a lead that this was a comedy film first and foremost and that they should expect the film to be unafraid of poking a little fun at Spidey.

Phil Lord: You may notice a bunch of scenes that are reminiscent of other iconic spider-men and moments. This version of Peter is
supposed to be an amalgam of all the spider-men that we knew in the
universe. Good and [pauses while we watch Spider-Man dance] bad.

Chris Miller: Good and great!.

PL: Sorry, good and great. That joke saved the movie.

CM: Which, the popsicle or the 'dance joke'?

PL: The dance move.

CM: That joke started the movie.

PL: I resisted that dance joke and Rodney [Rothman] pushed hard for it [both laugh]. And he was right... It told the audience what movie we were
watching. That they were watching a comedy. They laughed so big and then
they laughed at everything afterward. As a result.

CM: I call that a warm-up laugh, Phil. Warming the audience up to laugh.

_{^{Into the Spider-verse: Makers Audio Commentary.}}