I can understand, but I dunno. It's one of those things that the game generally glosses over for PCs. If you're going to concern yourself with that kind of issue, you get into the really nasty aspects of 'How modern IS this fantasy world?'
On the one hand, I'd say no, don't worry about it, because it's not something we worry about for PCs, so why is it a concern for animal companions?
On the other hand, I'd say yes, because if this IS a topic being handled maturely, it is a possible issue, especially for tracking and such. (What innkeeper in their right mind would allow a 300+ pound animal in anyway?) Some inns I've run have a policy about keeping animals outside, perhaps in a stable/kennel. Training the animal otherwise might allow you to keep them nearby, or at least pushing the issue becomes a good side-issue for RP.
And on the third Athach's hand, there is the 'dung-ages' motif, where we invoke this quote (paraphrased): 'How do you know he's a king? Because he hasn't got shit on him.'