It seems like it should have been illegal that Lenny and Carl are so far apart in their appearances, but it is what it is. Carl appears at this stag party, with the "wrong" voice.

For the most part, I would say Carl's definition within the show is similar to Lenny's, he is often used as an everyman, an average guy for Homer to hang out with. The main things that sets him apart (for better or for worse) is the fact that he is black - probably the most prominent black character in the show. Thankfully jokes about Carl's race are relatively tame, especially considering that things like South Park and Family Guy would attempt much "edgier" race humour in the show's lifetime.
The other defining feature of Carl is that he's often more straightforward and no-nonsense than other "average guy" characters. He'll joke around with friends, but also be oblivious to their feelings, leading to one of the few absolute fourth wall breaks in the show, in season 11's "Pygmoelian":

Lenny and Carl's unique relationship is formed by the tropes of living within a sitcom - they are almost always on screen together as a sounding board for Homer, with little existence beyond that, at least at first. The show, savvy enough to acknowledge the absurdities of a regularly scripted sitcom format, takes advantage of this and extends it to a logical conclusion, that these two become treated as a singular entity.
Most of the time their relationship is treated as a close platonic friendship, but the more this goes on, the more they are portrayed with homoerotic subtext. It's complicated - their existence as a potential gay couple is played for laughs, as if it would be silly that these two men would actually kiss. The show skirts right up against the line for this, without ever actually confirming them having sexual attraction towards one another. Is this queerbaiting, or is queerbaiting only if audiences actually care about if the couple actually get together? Because I doubt many are actually desperate for these two to be confirmed as an item.
(Disclaimer: yes, I'm sure certain parts of the internet are very heavily invested in this. I'm talking broadly.)
A minor aspect of this is that often Lenny and Carl's relationship appears to be relatively one sided. Carl is visibly annoyed when Lenny is partnered up with him in "Mountain of Madness". Lenny carved Carl's face into a mountain as shown in "Half-Decent Proposal", but Carl has never acknowledged it. Lenny things of Carl as his soulmate, but Carl does not reciprocate. Maybe this relationship is more tragic than I thought.
A fun aspect of Carl is that he has Icelandic origins. This is presumably a by-product of his surname being the Scandinavian sounding "Carlson", which is simply a by-product of the writers thinking it would be funny if Lenny/Carl had similar forenames and surnames. This kind of bizarre character detail and evolution is unique to The Simpsons, where 30+ years of refinement and exploration are possible.
This post is part of my "Every Simpsons Character Ever" series. For a list of my rules in this project, click here.