I just finished the game with my 4th char, an Elementalist. I tamed a Phoenix, as this fits the Fire Elementalist theme very well.
Now they are quite large buggers, and I compared my lousy level 5 Phoenix to a Level 20 Hearty Phoenix.
Result: Absolutely same size. 3 people could not see any difference in size.
I also believe this is also true for level 15 and level 3/5 Tigers.
Tyrian pets grow in size as they evolve, but Canthan pets do NOT seem to grow at all.
Can someone check/confirm this? I can offer Tigers, a Black Moa and a low level Phoenix for size comparison purposes.