DM is not comfortable with your reskinning of an Owl familiar. You have several choices:
leave the game and play elsewhere,
use a regular familiar (owl) and use your imagination as to why your familiar is an owl (perhaps it originally was a dragonling, which got cursed or polymorphed or bound into a familiar form, so that it may only shift shape into those set familiar forms as in the PHB).
Choose Chain Pact and have a Faerie Dragon or Quasit or an Imp
Or in your imagination, your (public owl familiar) is a dragonling or Bat, in appearance. Very weak so can't lift or carry, has the stats of an owl and is called Draco or Bat.
At the end of the day this is a fantasy game of the imagination, it is meant to be fun for you, the DM and other players. Yes in an AL game there are stricter or looser rules, there is also an increased chance of playing with players new to the game, so a DM would naturally consider this and not want to confuse other people. A rose is still a rose, whatever you call it.
I personally would choose the Owl Familiar, call it Bat or my familiar. Then refer to it as Bat or my familiar giving it tasks it can do, others will imagine a small owl helping you whereas I would see my Bat familiar. Some Wizard or Warlock casts magical Darkness and my poor familiar would be under the effect of the magic, which causes its (Owl) Darkvision to fail.
A great DM would accept this, a good DM does not need to know.