Halo is just barely an example of what you're asking for. The galactic community sans Earth has been conquered by a religion that humans are too smart/strong to surrender to, but the Arbiter allies with Master Chief to defeat them.

The second half of Fire warrior has you fighting along side space marines.

Freespace 1 and 2, former enemies the Vasudans team up with the Terrans to fight the Shivans.
Wing Commander has a couple of races join the Confederacy though the best example is the defector Hobbes in Wing Commander 2 and 3.
Half-Life 2? At least there is a friendly Vertigon or what it was called.

X-Com Apoc has you able to recruit the Alien Hybrids, which are half Sectoid, half Human.

No Man's Sky

Also XCOM2, at least with War of the Chosen (don't remember before it)

Also Psi5 Trading Company

I agree with LegoDnD. The last few missions in Halo 2 and the missions in Halo 3 have you working with the Arbiter, a fraction of the covenants, and the flood even.

Advent Rising

This game is the original Mass Effect.

The last mission of Starcraft 1 lets you control both protoss and terran units, fighting a zerg AI. This is inverted in the last mission in Brood War, where you are a zerg player up against terran and protoss teams allied with each other. IIRC, Lego Batman 3 lets you play as Superman (the alien), who can buddy up with Batman (the earthling).

I'm not sure, what are you talking about. There are plenty of games where humanity allies itself with aliens.
StarCraft
Star Control
Starflight
Unreal
Half-life 2
UFO: Afterlight
Practically all space 4X strategies allow alliance with aliens (in Stellaris you can literally make aliens your citizens and even decide if laws would apply to them equally or not)

Any X4 game where they allow you to create a multi-racial empire (usually through conquest).

FTL.

Song of Saya, depending on who you are rooting for :P.

Streets of Rogue (mutliplayer with one of the players playing the alien character)

Freedom Force (and a lot of superhero games where at least some of the super powered characters come from other worlds)

Another World (buddy)

If you play Master of Orion, humans have improved diplomacy thus are the best at making friends with aliens. My pick would be to make friends with the Mrrshan (cat aliens) and the Psilon (four arms for extra hugs) and just go to war with the Darloks that are spying on everyone 24/7.

Starflight also has alien races where you can have them on your ship as members of your crew. The only limitation that I remember was that the reptile aliens and plant aliens outright hate each other to the extent that you couldn't have both as members of your crew. If you had either as part of your crew then aliens of the same species would be friendly, but the others would be hostile. You can't be friends with everyone in this game.

Homeworld, where throughout your journey, the Bentusi will assist you by trading you new technologies in exchange for the minerals you've mined from asteroids. That's the first game, as for the second one, I don't know since I've never played it before.

Check out literally any space 4X or space RTS game, they've been around since before the space opera games and you can ally with aliens, ally with aliens against other humans, ally with aliens against other aliens, ally as aliens with other aliens against humans, etc. There are plenty of them; Galactic Civilizations, Endless Space, Stellaris, Imperium Galactica, Master of Orion, even Starcraft.

Septerra Core and probably any other JRPG set in space will have alien allies in your party.

And then of course there is Subverse, which can almost be considered the logical endpoint of Mass Effect, but its still in early access and the combat is Xcom-style and still unbalanced according to reviews so it might be best to wait for full release or a proper sale. It only has one dude and has plenty of cooperation between him and the aliens, if you really think about it.