In the case of your wish being granted, it is implied that an omniscient being is making your wish come true. In western culture that would be God or Jesus. It implies that someone or something heard the wish and decided to allow you to have whatever it was you wished for. In science, a grant is a monetary endorsement for research. Much like a wish coming true, a scientist receives a grant from a benefactor.

"Grant" is a fairly old fashion word for "give", that only really survives in the saying you mentioned; granting wishes, etc. You could think of "He granted my wish" as "He gave me my wish". So, if I "Grant your wish", I would be making it real. It isn't reserved for God-figures.