Did we ever have this function? Or am I imagining things? 

I was never on HP from 2009 - 2012 so if this was a thing, I am not aware of it, so feel free to correct me if I'm inaccurate. But, I think the farthest it ever went was discussions here in the forums as quite a few players have suggested it becoming a new feature, though never became legitimate.

Typically, many would state their "cheat" to doing next horse / next dog was opening up a line of tabs in their preferred browsers (Safari and Opera are two I remember coming up a lot) of all animals in that account and going through each one that way to make it a little faster.

Been here since 2009 and it's never been a feature here that I can recall. I don't know if I'd use it if it was? I do multiple tabs (1 divider at a time) when training and it works out perfect as each divider is trained differently.

What Con said. 

I can also say that I've noticed my animals shuffle around within their dividers so I feel like the next feature would make me miss some while training and that would keep me from using it and just keep using LinkClump in Chrome to open up an entire divider at one time.

OMG Siib, I'm so glad you said something about the shuffling. I thought I was going crazy xD 

The discussion must have been what I was thinking of. I vaguely remember it.
Opening tabs is what I do when training, and I noticed the shuffling recently. Usually I number my dogs within a divider but recently I've decided to wait and see what the points are like at breeding times so I can pair up by points.
Thanks for the replies!

i had no clue about next but i have a gold fish brain basically XD
i just tab everything out 

I tab mine out like said above - but Firefox makes training sooo much easier than Chrome. I am regretting my Chromebook lol

I tab everything, too, and I don't wanna change that xD however, I've used Firefox and Chrome for training and I'm fine with either one, but I do prefer Firefox. Just gotta figure out what's wrong with mine...