I only get the "Begin animal training" with fresh tames, not with my old ones?
No. It works also with old ones. Even with bonded pets.
I think u tried to train a pet that already uses 5/5 pet Slots.
I can train all my pets that have 1-4 pet slots. My Greater Dragons that use 5 pet slots cant be trained (old and new ones)
U can find the official Infos here:
"Animal training is an advancement to the traditional pet skill training through which a pet has, in the past, gained its skills. The training can be undertaken with any tamed creature be it fresh tamed or ‘fully trained’, ie with skills at maximum. The exception being pets requiring 5 follower slots, these are considered to be already at the maximum power and therefore not eligible for Animal Training."