Can i get my horse into a boat?

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!

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.

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.

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+