LK/MK chains just get V-skilled by Gief. I'm not sure that's intended behavior for a hit grab, but it's how it works right now.I've played it from both. It's the same as with Hawk, you want to keep Gief out because he benefits more inside than you do (more range, better damage on SPD). Birdie has more tools to keep Zangief out than a ton of chars in the game, lol. Well-placed MK or HK chains, st.mp into rush punish, cr.lk confirms into rush, command throw, etc. Then you have V-Skill and his buttons as well.
Mika has a way harder time in this match because her buttons are hella short and she has fewer methods of keeping him out.
I was comparing Birdie's (horrible) Mika matchup to Gief here.