She can't say her name or write anything but Blake does magically understand anything else she says to him and she did tell him her age indirectly, and as far as I know ser doesn't know a thing about B and A's past so can't she just talk about their memories together or show interest in the things she did before in front of B? Or like cook Korean foods like she did before ? Or straight up say her sister's name or tell B to go to her sis together, I mean there's a whole lot she can do to show B and everyone else she is the actual A. It's not like a person's identity is their body, it's memories, body language, the behavior, likes and dislikes and so on... but somehow she just tries telling him her name like it's all there's to her identity ( ̄へ ̄)