That's obviously true, but I don't think it makes any difference to the question you posed - since whether or not a B22 bulb/'lamp' should or should not bear a Class II marking is a 'standalone' question which has nothing to do with what it is (or may be) plugged into.If all BA22d lamp holders were Class II then there would be no problem, however I have selected this example Clearly some bulb holders do have a connection for the earth as seen ...
... by analogy, say, a Class II 'walwart' should have the Class II marking (relating to itself), regardless of whether it is plugged into a metal or plastic socket.
Kind Regards, John