I read this thread at Minecraftforum.net
and decided to test it.
I built a Nether portal (in Creative mode), spawned in a horse and pushed it into the portal, it just stood there without porting.
I tested with a bunch of other mobs including several horses, all of the others ported normally.
I used /data get entity >UUID< to check the Portalcooldown:, when it was standing in the portal the cooldown was at 300, when it was outside the portal it counted down to 0.
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Could be, I have a Drowned on another world that has been standing in a portal for quite a while now without porting.
(Making a backup in case the data contains anything useful (though I haven't found any way to copy and paste it yet.))
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This may well be normal behavior though, I have a water stream on one side of the portal to push mobs in, so if they port to the Nether and back and then try to leave the portal on that side they would be unable to leave until they turn around and go the other way.
But there was no water involved in the world with the horse and I observed the cooldown timer counting down to 0 before the horse wandered back into the portal and still didn't port.

I initially pushed the original horse into the portal but later it was wandering around freely going in and out of the portal without porting even when the cooldown timer reached 0 before it went back in.

Entities (except players) cannot go to the nether if no portal exists on the nether side

@Juev That may indeed be the cause of the problem but the problem persists after I have used the portal, newly spawned mobs go through but the ones that were pushed into the portal before there was one at the other end still won't port, even after wandering away for long enough for the cooldown timer to expire and then wandering back into the portal.

I've been looking for a remedy to this, problem persists in 1.14.3

Also persists in 1.14.4
Not that I was expecting any difference but taking the horse to a new (already used) portal that connects to a new portal in the Nether doesn't help.

I've done some more testing, and for some reason I can't recreate the bug in a creative world. Also I misread the post, I was able to get my horse into the nether, I just can't get him out.
Thank you for your report!
We're actually already tracking this issue in MC-151648, so I resolved and linked this ticket as a duplicate.
If you would like to add a vote and any extra information to the main ticket it would be appreciated.
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Sounds a bit like MC-151648