The Chapala airport is a private club. The airplanes there are ultralights or homebuilts.
https://www.facebook.com/aeroclubchapala/
For an ultralight license you must be Mexican by birth.
https://www.gob.mx/tramites/ficha/expedicion-de-licencia-para-piloto-privado-de-aeronaves-ultraligeras/SCT1692
If you want to explore getting a private pilot license , understand it is very expensive to complete the requirements.
There is a school in Guadalajara for getting your license.
https://www.facebook.com/ALTUSFLIGHT/
For a private pilot license you must be a Mexican by birth.
https://www.gob.mx/tramites/ficha/expedicion-de-licencias-para-el-personal-tecnico-aeronautico/SCT1676
If you have an license from the USA with a current medical, and bi-annual then it is valid in Mexico.