How to emulate an RFID tag with the RC522
No. Looking at the datasheet, card emulation is not a feature provided by the RC522.

Another popular chip, the PN532, does seem to indicate support for that feature, however, looking at the most popular library for this chip from Adafruit, it does not implement this feature. Another library from Seeed, does seem to implement it, but it looks rather limited as to which cards can be emulated, and the library itself hasn't been updated in a while.