Can i get my horse into a boat?

I recommend you use a lead to tie your horse and ride the boat like in this picture:

Don't worry about the lead, as long as you are in the boat.

I don't want to give out misinformation as I don't have any proof that Mojang intentionally removed mobs entering boats, so I've removed the update section of this answer.

Additional proof that this is, in fact, replicable in 1.16+

Make the boat stuck so it doesn’t move then push it into the boat. Best if you trap boat in a 2x2 and walk onto the boat with the lead to the other side of where the horse is to lead it in.

Get them close to the boat and walk into them repeatedly to 'push' them towards it. It might be easier if you use a lead on them and guide them directly into it, though.

You can get both inside.

1. Find/breed a baby horse/donkey

2. Push it into the boat

3. Stuff it with apples to grow

4. Tame it

5. Put a chest and/or a saddle on it

6. Enjoy your ride!