First a bit about my group's tendency: we choose our character race based on stat boosts and whatever ability the race might grant. After that, race hardly matters at all. And, I'm trying to think of ways to make it matter.
I am not sure about whether your group tendency is because they
like playing that way, or because they think they
have to.
I am going to assume that you and your group are together with the idea of making race matters more. But if they
like exploiting racial benefits, then you're going to have to offer more of them, while if they don't necessarily like it, you can make races matter more only in terms of story and narrative.
Classes seem to get all the love, but what mechanics aside from a few racial feats that don't really "distinguish" characters IMO does race get in the game?
Classes get more love because a class defines the
functional role of the character within the adventuring group. Then, both a class and a race (and a background) can deliver the
narrative role.
That's perhaps the reason why in 5e both races and backgrounds don't normally provide
functional benefits beyond the first level. Of course, in older editions such as BECMI the playable races were classes, so this just proves that there isn't only one way it has to be. But the current situation in 5e is that indeed the only racial benefits you can acquire
during a campaign are in the form of very few feats from a supplement, or possibly some race-restricted magic items (I don't even remember if there are any in the DMG, or if you have to design them yourself). No other mechanics officially provided.
I can't make my characters role play. I can't force them to play the differences between an elf vs. a dwarf.
That's ok, I don't think it's generally a good idea because it leads to stereotypes. Making being an elf/dwarf matter is not the same as making all elf/dwarf be stereotypical.
How do others handle the distinctions between race? Do you also just ignore it?
...
So, I ask a genuine question: what do you all do to make race matter in your games?
I don't. I have DMed 5e for 3 gaming groups, plus a PbP game. In the PbP game I let the players play races normally, but in all my tabletop games I did something different. However the main motivator was that I always had more beginners/casual players than people who actually played D&D before, and I just noticed that
at first level the amount of stuff that a typical race (except Human) grants roughly doubles the character complexity... because of this, I have always either
ignored racial benefits (i.e. let players pick a race for narrative reasons, but everyone uses Human stats), or
simplified them to 1-2 benefits instead of the full package.
So what could I suggest you to do in your case...
Functionally there really is very little you can do with the official material. As I mentioned, there are only Xanathar's racial feats, but they are really few. IIRC each character would ever only qualify for 1 or maybe 2 feats, so there is almost no choice, you either take
the additional feat for you, or you don't.
But generally speaking
feats are a very good idea, because they give the player a choice, and they are already built-in into the class advancement rules. You don't need house rules to design new feats. So if you're into some homebrewing, I suggest you design a good bunch of feats for each of the races that your players are currently playing. It's also probable that some 3rd party publishers already made this, if you'd rather buy some material than design it yourself, or look around this forums for shared material.
Beyond feats, you could introduce racial backgrounds (not good, they are again a 1st level choice, and of minimal impact), racial subclasses (much harder to design) and even racial equipment (not just magic items with a label "elves only", but also mundane items that only a specific race is proficient with).
edit: There's another "trick" but I don't think it works in your case, since you've already started playing and your players probably already have the PHB... but when you play with beginners who don't even have the PHB or know the game in advance, you can always
present character options as race-exclusive (or also class-exclusive, but that doesn't matter here), even if by default they are open to everyone. So you could for example say that the Assassin subclass is half-orc only, that illusion spells are gnome-only, or that the Magic Initiate feat is elf-only. If instead of ~80 feats open to everyone you had maybe 10-12 open, and all the remaining feats split up into the different races, it definitely deliver the
feel that race matters more!
Narratively instead, the sky is the limit... you could start by allocating territories and nations to specific races in your fantasy world, instead of the usual cosmopolitan standard, and then strengthen the ties of all members of each race to their kingdom and culture, for example in terms of call of duties, bonds to NPCs, seasonal affairs etc.