My T, after 9 years, got my name wrong :-) so I know she had no clue when my birthday was. I can't imagine a T knowing my birthday anymore than a doctor or other professional I employ. I don't think they read official papers or anything; may note an "age" but not without prompting, etc. We once discussed ages (her asking me to guess how old she was) and occasionally she'd say something like, "you're X years old. . ." but she was more or less quessing at/trying to remember my approximate age and I usually had to correct her.