Nope, you can only domesticate one animal at a time. Takes about 10 minutes (20 if you sick) and success and failure is about 60-30 in favor of failure.
And you need to domesticate every animal 10 times successfully, too. Guess that is the reason the person that abandoned my first animal husbandry villager with a full stable.
I've got 5 animals trained up completely by now. and it takes a time. :}

Holy crap x.x

Yeah, and even with all of this, I still want one of every pet trained. :}

Naturally

-running own seeds cuz running low on necessary plants for cures-

I'm trying to remember who my second herbalist is, I'm trying to figure out if its an option to have them do the plants, so that Robyn can do the exploring, and Starling can see to blacksmith work. :}

edit; apparently they have implemented a villager cap. As I can't create a new slot for a villager anymore.

How many villagers are you at?

Also 43k
Yowza

I have 10 and was looking to get an 11th just to have a second blacksmith.
I'm finally starting to save money now that I'm no longer spending nearly every penny I have on new weapons.
Hence the reason I wanted a second blacksmith. :}

Dang
Guess I can't have a second explorer

Hough I could've sworn that I'd seen someone mention getting their 12th villager somewhere...

I know, but if they like set a cap of say 7, this morning, anybody who already had more villagers is allowed to keep them.
So I don't know what the cap is, I only know there is one. :}

I can still build at least one more house, so the max is at least nine
I wouldn't be surprised for it to be ten, so someone could have one villager in each class