According to the Elemental Evil Player's Companion, a player can be an Aarakocra.

According to the Monster Manual (p. 12), five Aarakocra can perform a ritual and summon an air elemental:

Summoning Air Elementals

Five aarakocra within 30 feet of each other can magically summon an air elemental. Each of the five must use its action and movement on three consecutive turns to perform an aerial dance and must maintain concentration while doing so (as if concentrating on a spell). When all five have finished their third turn of the dance, the elemental appears in an unoccupied space within 60 feet of them. It is friendly toward them and obeys their spoken commands. It remains for 1 hour, until it or all its summoners die, or until any of its summoners dismisses it as a bonus action. A summoner can't perform the dance again until it finishes a short rest. When the elemental returns to the Elemental Plane of Air, any aarakocra within 5 feet of it can return with it.

I was wondering if five Aarakocras PC could summon it RAW/RAI. In no place is it specifically said that a PC can't summon it, nor says they can do it.

In case the answer is no, I was wondering if it would be balance-breaking to allow it. What unexpected side effect could that produce?

Short answer: No, by RAW

The monsters and NPCs in the monster manual (and equivalent sections in other supplements) do not necessarily follow the same rules as player characters.

This can be seen just by looking at the various human, elf etc NPCs in the monster manual and noting that many of them have differing abilities to those noted for their equivalent player races in the PHB.

Similarly, there are a few monstrous races such as the hobgoblin, goblin and so on presented in Volo's guide, specifically presented as player races. But these do not follow the same exact rules as their monster entries in the Monster Manual (even if they keep to the same basic 'theme').

As to whether it would be unbalancing...that rather depends on level. It is a powerful ability. It is slightly offset by the length of time is required for the summoning, but this generally wouldn't be a big deal because it lasts for an hour and the players could simply summon the elemental prior to entering a dungeon or whatever.

Having a CR 5 creature under your control would be very powerful for a low-level party. For comparison, Conjure Elemental is a 5th level spell (typically accessible by a 9th-level spell-caster).

RAW, Aarakocra PCs cannot do this

As you've pointed out, there are no rules that say that 5 Aarakocra PCs can or cannot do this, and only the sidebar in the Monster Manual suggests that 5 Aarakocra can do it at all.

However, the rules for "monster" Aarakocra have no bearing on the rules for PCs, so this is not evidence that Aarakocra PCs can do this. There are also no rules that say that 5 Tiefling PCs can get together and summon a devil, but no-one believe this is possible outside of DM fiat.

Conjure elemental is the only option

If you wanted to make an Aarakocra PC that can summon Air Elementals, then finding way to cast conjure elemental is the only RAW option that I know of, such as learning it via being a spellcaster such as a Wizard or via the Warlock invocation that let's you cast that spell. It is something you can then do by yourself, so it isn't quite the same as having 5 Aarakocra get together as per the Monster Manual sidebar, but it's the only way RAW.