You can say that a neural representation of abstract objects physically exist; that wouldn't be controversial for a physicalist.
You need only postulate, however, that certain computational systems (like our brains) recognize similarities between distinct objects; this seems rather weaker than what one normally means by an object (abstract or otherwise) existing. For instance, one could theoretically remember every case of an item that was called a "chair" and perform a pairwise comparison between a new object that may be a chair and all of those; and if this new object is as similar to the other chairs as they are to each other, and there is no other label with a set of exemplars that is a better mtach to the object you've got, then you have a chair. This is a perfectly functional system for naming and recognizing chairs, but what in this description is this "abstract chair"?
So it is not necessary to suppose that abstract objects exist, even if we act as though there is such an abstract object. Ultimately, a physicalist probably ought to defer to evidence regarding whether abstract objects effectively exist in brains or not (the physical world certainly contains regularities, but one is under no obligation to create an abstraction to instantiate those regularities).