Follow Up Question: Are there other exceptions for when być (and its various conjugations) do not lead to a instrumental object or is it only in "This is X" sentences? For instance, one does use the instrumental in "Y is X" sentences so long as jest is used instead of to, correct? (Also, thank you for always having such speedy and helpful replies, Jellei.)
You took it a bit too far, "wasz" is just a possessive pronoun. Actually possession works more or less like in English, you either have "your dog", or, let's say "Adam's dog" - and this construction is called Saxon Genitive.
So while "your dog" is simply "[twój/wasz] pies", "Adam's dog" will be "pies Adama", putting Adam in Genitive. Oh, yes, it's not the 'owned thing' (pies) that takes Genitive, but the owner.