Is there any royal family in Mexico or royalty of some ancient Aztec culture or something?

No, as the case with most "New World" countries, the political system chosen by the people or those in power has been one of revolving leadership where the powers that be leave power on a predefined basis. Mexico is a democracy and as far as someone from the Aztec Culture being a descendant of a King, there may be such individuals, but any recognition of their power as King or any political influence is non existent. Hope this helps.