Why does one seem better than the other to you?
be means "equals." Have means "possess" as in possessing a quality. You could also write I show Scottishness, I have a Scot nature, I show a Scot's mannerisms, I have the viewpoints of a Scot. You could identify your nature more explicitly by stating what characteristic makes you seem Scottish: I talk/sound/look/dress/smell/eat/drink/swear/snore/taste/spend like a Scot. Frankly, I would find any such expression more informative and satisfying than I
am Scottish or I have Scottishness.
I would find it entirely correct to hear you say "I
am a Scot," provided you had Scottish-born parents who bore and birthed you in Scotland.