I've played most every version of this game, PS1, Segasaturn import, Psp...360 now and in the 360 version the Bat Familiar is seemingly glitched its slow as hell never keeps up with you, I've literally speed ran from one area to the next and it took over 5 minutes of waiting for the bat to catch up, It catches up instantly when you go to new areas but in any given loaded area it can't keep up it literally does nothing unless I stand still and keep it in my general area.
 
the rest of the familiars seem to be fine, the bat is too except it doesn't keep up with you like it did in the other versions of the game.

" @WilliamRLBaker: Yep just tested it and it's the same here. I did have to download a "title update" before I played though so I guess it's possible that it's a result of that. Or maybe it was just like that the whole time. I never played much with the bat so idk. Crappy either way though. "

cleared cache, its not the title update that did it.
Likely is just a product of the game being ported.

[XBLA version] As far as I can tell, bat familiar isn't glitched. No matter where you are, switch into bat form and the bat familiar (if active) will return to you within a few seconds, and get on with his task of smashing any enemies in sight. Looks to me like a deliberate decision on the part of the game designers. Hope this works for PC/PS3 users!

As someone who played a crapload of Symphony of the Night on the PS1, and used the bat as my primary familiar, I can confirm that this is most definitely a glitch. No matter how fast you're going, in any form, the bat will keep up.