Clarification: you can feed them even if they are at 100%, force feed them basically. It has never caused any problems for me and it does unlock skills for training. It's almost certainly the likes since the are 100% energy long before the likes stop increasing at 26.
You can also increase likes by naming the pets again and again, same name is fine, you can get to the max (50 likes) doing this, again it has never caused me any problems.