I thought it looked something like a berry vine too, we have wild raspberries growing in the back it's not quite like that, but similar. I know some of them can be toxic if they eat enough of it.
My rule for my dog is, if I won't chew on it, or put it in my mouth then she shouldn't either. You never know what it could do to them until it's too late........ I wouldn't allow him to continue to eat the plant until someone identifies it, and then still, most plants are toxic in some form or another if they eat enough of it.