OOoooo!! While I do like to show off mounts sometimes, I more love to match mounts to my character/transmog or use them thematically in other ways. It’s not often I use the same mount for multiple characters at the same time, but some mounts have been used by multiple characters at different times. I most often swap around mounts I use at expansion changeovers. Another thing is I use sets of mounts for my characters. Each has a flying mount and a second mount, which frequently is a preferred ground-only mount.
Some examples:
– This expansion I picked mounts which match my character’s chosen covenants (though not any of the covenant ones). Examples, my Night Fae hunter rides the Vulpine Familiar… my venthyr demon hunter rides the red/black Prestigious Midnight Courser with it’s fancy armor.
– When Wrath was announced, I farmed the Wintersaber mount to have a snowy white mount to go with snow. Then there was less snow than I was expecting and I had to make a last minute mount change. 
– My shammie is my only character which changes spec (resto/elemental) and I use her mounts to help remind me which spec I have turned on. Each spec is set to different mounts.
– A few mounts feel “owned” by the characters who got them and are near-never ridden by other characters. For example, in a very unlikely turn of events, it was my shammie alt (not my main!) who first scored AotC for SoO and thus my shammie is the near exclusive rider of the Kor’kron War Wolf among my characters.
As far as problems with mounts go, there are several. I tend to mounts that are smaller and won’t become stuck on things. There are several large mounts I adore, but their size prevents me from riding them regularly, most notably, the Dreadwake and the cloud serpents. Also anything which doesn’t touch ground to land but also can’t fly is a problem because if I’m trying to jump around thin ledges or small surfaces, I need to know where exactly the mount is touching. Especially when in a no-fly zone, this includes all phoenixes (MoP challenge, A’lar, Dark), again the cloud serpents, the rogue crow class mounts, etc. I’m not a huge fan of anything which doesn’t LAND when not moving and technically “in contact” with the ground.
Fun fact, Sinrunner Blanchy is the first mount I can remember shaping up to be a mount I use a lot and with multiple characters at the same time! 
Even now we can fly in Shadowlands, Blanchy is set as the “second” mount for a lot of my characters. Go, Blanchy!
