Isn't κάρτα also a postcard?
Well, κάρτα all by itself doesn't necessarily refer to a postcard. If you'd like to do so, I'd suggest that you use ταχυδρομική κάρτα or καρτ ποστάλ, just to be a bit more specific. ^.^
Would " η δική μου" also be correct?
I think yes, but it would mean "this card is (the one that is) mine, contrary to the others".