there is nothing that is printed on the SoC.
That sounds like not a real NXP MFRC522 chip, so this can be anything and I doubt that a software change will activate the other protocols.
I thought once a tag conforms to the MIFARE 13.6Mhz standard then all of them should behave same. No ??
There is not one MIFARE standard but that's a bundle. The NXP chip supports MIFARE Mini, MIFARE 1K,
MIFARE 4K, MIFARE Ultralight, MIFARE DESFire EV1 and MIFARE Plus RF. Maybe your chip does only support one of them. As your supplier :-).
As time passes you realize that E-Bay is a good starting point for makers but not for serious industrial solutions.
Wrong, it's a good starting point for bargain hunters that don't care to throw the thing away if it doesn't work. Makers usually need at least a basic documentation but on ebay you usually get nothing. So in the end it's a good starting point for frustration and hassle. If you really know what you're doing and what exactly you need it's better to use AliExpress as there you have a higher chance to get in contact with the manufacturer which might provide some documents, even if they are in Chinese.