Indeed it appears to be dual frequency and according to the manufacturers specifications it reads either MiFare (NFC) or EM40xx (LF) as Standard. It can also read several other formats optionally.
Given that you can’t scan it with taginfo we can assume that it is low frequency and an Xem or any one of a number of chips (like the NeXT) can replace it.
As for payment I am assuming that you are in Hungary as you speak Hungarian so a WalletMor is currently your best bet.
In order to copy your access card you will either need a blue cloner (but they are not recommended) or access to something like a Proxmark3 easy.
It is probably a good idea to buy a card or two to copy your access card to first to test that you can copy it successfully.
I don’t know if anyone else here is in Hungary, but of so they might be able to help with the cloning.