I'm grinding this now, and it's important to realize that not every exhibit animal has a variant. I set up a breeding zoo in the sandbox, where all breeding research is maxed out, and after more than a dozen attempts, I kept getting no variant mountain bongo antelope babies. I go to check the albums and discovered that there is not even an empty slot for a variant mountain bongo antelope. So before you make a futile attempt to breed a species variant, check the albums to make sure that one even exists.
Update: It seems like the only breeding species without a variant are the Arabian oryx antelope, mountain bongo antelope, sloth bear, and sun bear. Hopefully, I didn't miss one.
I systematically ground through this using one of the useful spreadsheets posted on this site. When I got what I thought was my last variant, the achievement didn't pop. My tracker said I was at 98%, even though everything was checked off and all variants were photographed. What could have gone wrong?
1. I read on here about the albino polar bear being a source of difficulty because of it being photographed in a campaign scenario, so I had already bred it multiple times. It was not the one I was missing.
2. There were other opinions saying that both the original species AND the variant had to be bred. I don't believe that is the case. When I was breeding the South American animals and I came down to the last animal I needed, the first baby that was born was a variant. Right after the birth, I got BOTH achievements (breed all South American animals and breed all South American animals and their variants). So breeding the variant alone should have counted "two-for-one." This hopefully could not have been the problem.
3. The only other possibility is could another variant have been born and photographed in a campaign scenario, and I didn't keep track of it. I went back into every sandbox breeding zoo I had set up and documented all the species variants present. Then I went into my big challenge zoo (the one that requires collecting 176 animals) where I had to breed 100 animals and release 100 into the wild to get the Zoo Tycoon achievement and documented all the variants that were still present. That left me with about 10 animal species unaccounted for (that probably got released into the wild), so I set up another breeding zoo to re-breed these animals. Near the end of this effort, POP! Got the achievement after getting the variant south central black rhino. I have no recollection if this variant was born in a campaign scenario, but it must have been. There's no other logical explanation.
The moral of the story: WRITE DOWN any animal that is bred and photographed during a campaign scenario because that breeding will not count towards the achievement. Only breeding that takes place in a challenge or sandbox zoo counts. It's too bad that the zoopedia doesn't keep track of this or variants in general.