However, the above create table query reports syntax error:
ERROR: syntax error at or near "."
LINE 10: UNIQUE (google_user.email)
Is there a way to create unique constraint on a field inside composite type?
Not tested here but in most (all?) cases when attempting to de-reference a component of a composite typed column you must place the column name within parentheses.
(google_user).email
Otherwise the system is thinking that "google_user" is a schema and email is a column.