Like so many ordinary and common words, cult has a specific meaning in the Catholic Church which sounds weird from the outside.
When a saint is declared a Servant of God by a Bishop, it gives the ok for a local cult to rise up around that holy person. This cult can extend to the entire world once the person is declared venerable.
So, in this case, the cult is the group of people who is asking for the intercession of a particular would-be saint.
If a saint's origin is declared dubious, as some have been in the last century, we say their cult has been supressed.