In Robert Asprin’s “Phule” series, there are two distinct cultures of Slug-beings. Captain Phule had initially made the mistake of treating his two Slugs as if they made natural companions for one another, until someone educated him that one was working-class and one was an aristocrat and they pretty much hated one another.
You might be interested to know that a lack of diversity in alien races is given as a textbook example of outgroup homogeneity bias in social psychology. (That bias being in the mind of the author).
The Tenctonese in Alien Nation had at least three different religions: the majority follow Celine and Andarko; Cathy Frankel’s was never named but she stated it was “more Eastern” to Matt Sykes and was clear she wasn’t a Celinist (and she was a vegetarian, as opposed to the omnivorous majority); and Uncle Moodri followed the Goddess Iona, a “pre-Celinist” religion.
The Tenctonese also had an underclass called “Eenos” which were culturally separate from the rest (basically, untouchables forced to be cannibals to survive while aboard the ship)
Good point. I read an amusing take on that somewhere - aliens hearing that Americans were good at coal mining, and picking up an assortment of Americans to work for them - assuming any Americans would do.
The hwarath in Eleanor Arenson’s Ring of Swords have visibly different races. One of the hwarath males we meet in the story comes from a northern island region and has thick white fur with grey spots (like a snow leopard) as opposed to the shorter uniformly grey fur that’s more commonly seen in the species; we’re told it’s a common characteristic of that particular racial subgroup. And the hwarath speak several different languages on their planet.
Including atheists, such as Na’Toth. And I think they did actually have different races, too, distinguished by the patterns of their spots (what, you think all species’ races would differ in the same characteristics as humans?).
While we’re on Babylon 5*, Lenier mentions in one episode that there are dozens of different Mimbari languages.
There’s also an afterword (purportedly a letter from the protagonist to Lewis about the details that he left out of the book) that mentions racial variations within those three species, too.
Jack Vance’s Planet of Adventure series.
Vernor Vinge’s Children of the Sky - the Tines who live in the equatorial regions have a dramatically different culture from the Tines who live in the colder regions.
The most obvious example in regards to Babylon 5 would be the the three Minbari castes. It’s pretty plain that they are culturally distinct from each other, and just plain think in different ways as often as not.
The Drazi have the whole Green and Purple thing, but that’s a weird example, and only becomes relevant in election years. 
There was a season 5 episode of B5 that dealt with one of the Alliance’s member races having suppressed information that they were related to the other indigenous intelligent life on their planet, who they had considered inferior before they had wiped them out. Turns out, without that other group on their planet, they ended up lacking some vital genetic lego piece and their race was slowly dying out.
In Mass Effect, you have the “Pureblood” Asari and then you have the Asari who intermarry with other races. Interestingly enough, it’s the ones that don’t intermingle that are looked at askance by Asari society as a whole. There is also mention of various Turian clans that, until the relatively recent Unification Wars, tended to fight it out amongst each other. Ditto for the Krogan clans, which was only starting to resolve itself by Mass Effect 2 (haven’t played 3 yet, taking forever for it to get to me in the mail…)
Oh, and the biggest Mass Effect example as of ME2 would be the Truth Geth and the Heretic Geth
In Harry Turtledove’s “Balance” series, the invading aliens are both multiple races and thrown off by Earth’s cultural diversity; he gets a lot of mileage out of that. They’re multiple races because one of them conquered the other two (each had originated in a different planet), but the conquerors hadn’t run into “such a bloody mess of a planet!” before. The two dominated races have been integrated by their invaders’ culture in a stratified way that doesn’t get explained in detail in the books I’ve read, all that’s clear is that there is a lot of jobs they’ll never have access to; the invading ships carry only members of the invader race but the slower creche ships are carrying all three.
In John E Stith’s Deep Quarry:
It turns out that the psychotically warlike species that built the ancient spacecraft that is discovered in the course of the story is just a different cultural branch of the pacifistic Womper race, one which they never admit to other species ever existed.
In Alan Dean Foster’s Tran-ky-ky trilogy, you have the medievalesque "mainstream Tran, the nomadic barbarian Tran, and the “Golden Saia” who are physically different as well as culturally (not genetically though; its a natural change any Tran’s offspring undergoes in a warm climate).
In Mother of Demons, the Gukuy have at least three very different cultures that come up in the story. There’s also a division over who does and does not follow this new and spreading religion/philosophy.