Okay so as someone who used to farm like mad and now still semi-madly farms mounts I have to ask who else does it then never uses them lol
Maybe it’s just the hunt or the journey being more important, as I have had a lot of sick fun farming Invincible, mims head, and so on but I rarely use them for longer than a day or two
I did the rare spawn mounts in Wrath and Cata. Boss drop mounts I’ve attempted a handful of times, and not all of them.
The process of farming them in the world was important. Competition with other players. It was a PVP server so there is that aspect. And I still use those mounts. So some of both I guess.
When I get a new mount I’ll frequently use it for a while, until the next shiny thing comes along.
There are several mounts I keep as favorites for various reasons. But, yeah, most mounts get phased out for me when there is a cool replacement.
Some are bragging rights, some are utility, some go with a RP theme, some go with a mog. The same reason I collect all the things… because they are there.
Number goes up, small amount of dopamine is released, am happy
Then it’s back to normal 
Up until this expansion, I mained DH, and my main alt was a warlock. So I farmed mounts for both the Felfire Hawk and Feltalon achievement mounts because I thought my characters would look cool riding them.
Mount++
The only mount I’ve honestly farmed was the Blue Proto Drake. I really “need” to farm the Green Proto Drake. I would like the Red, Blue, and Green Proto drake to be the respective mounts for my DK. I don’t play Unholy a lot, so I don’t miss the Green one that much. But I’d like to get it (those dailies are just annoying…)
Arguably I also farmed the Firelands Hippogryph.
I like Hippogryphs, I have all of the (save for the Refer a friend one). But past couple of expansion I’ve been going with themed mounts (Korthian mounts, DF mounts, I’m using the Rook mount right now).
In general, if a mount is low hanging fruit, I’ll grab it. I bought the Kitteh! mount (and used it for most of SL).
Otherwise, I’m not going to fixate on it.
I MAY buy the AH mount, still undecided on that.
I haven’t gone out of my way for a mount since the Vanilla Warlock epic mount quest line. I just kind of get the ones attached to the seasonal content I play or luck out on a drop here and there like I did with the GMOD.
I do get a bit excited about getting them when I do but I don’t go out of my way for them.
I farm mounts because I like options for themes. This character uses the magic rooster because it matches my theme. Before I was able to get this mount I used a goat mount. Having so many mounts means I can give my characters mounts that match them. If I don’t have that perfect mount yet I can find something that works until I can get it. Then sometimes I just like the look of a mount. I may never use it but a cool looking mount is great to have.
I guess for me, the answer is all of the above.
I want to use them.
The rush is also a factor.
Increasing my mount count is also a plus, since it is technically working toward something when they add a new achievement.
I have a lot of alts, and use the [Per Character Favorite Mounts] addon, so I can easily add any new mounts to those characters who will use it.
Lots of alts means lots of themes, so I have lots of mounts for each alt. Often I do farm/get some mounts because they are an ugly recolor and I just want my number to go up, but usually I grind for a mount I really want because I know it will go well with a theme of one or more of my alts.
I like to have options. Certain situations call for certain mounts. For instance, I like broom and rocket mounts for mining and herbing on non-Druids because there’s little to block your view of the ground and you can spot ore and herbs easier. In PvP I tend to use PvP mounts. In cities I use small ground mounts. When flying long distances I like a large majestic drake, and when afk’ing in Dalaran, nothing’s better to sit idling on outside the Alliance portal room than a Warlord’s Deathwheel. 
It’s usually for the count. On my druid, I rarely use actual mounts since I have travel form and mount form. On other characters, I rarely switch which mounts they use.
sunk cost fallacy. “It will definitely drop next time.” ~ me after 19 years.
Most of the time I farm a mount it’s for aesthetics and class/spec fantasy.
Like when I tank on my dk I like to use a “tankier” looking mount like the revendreth gargoyles or the mech suits.
In frost spec I love the spectral charger from Torghast.
On my monk I like the cloud mounts.
It’s about what matches the look and role in that moment.
I didn’t start as a mount collector but after one rare-ish drop, I got the itch and the hunt is fun. I use one or two mounts unless its a holiday or i get really excited about a mog. Its also been a fun way to spend time with someone, doing a random quest or returning to an old raid together. so I would have to say its for fun rather than numbers for me. 
Most I get because I want them for a specific character and transmog. Like this character…I really want the Corrupted Dreadwing from WoD but…ugh…all those Apexis Crystals…and it’s not a shared currency yet…if it were a % chance to drop mount I’d be all for it, but something about farming those crystals has me turned off.
Anyway, it depends. Some I farm to use on any and all characters and will use them for weeks or months at a time. Some I only want for specific characters and mogs. Some I go after as just a +1 because I’m within like 80 mounts of the 500 achievement and might as well tick that box while I’m at it cause I do want the achievement mount. I know some people who do everything you can possibly do in every expansion just to get the maximum amount of things they can get, mounts included. All reps maxed, etc. I don’t have the time or patience or I’d be well beyond 500 already. It’s mostly quality over quantity for me. I don’t have the most rare mounts in the game, but I’ve got quite a few that are rather highly sought after.