Why does Trish's Stand have a personality of her own?
These are known as Sentient Stands:


  Although most Stands do not show any personality or simple hints of it, there are Stands who display a full-fledged consciousness separate from their Users. These Stands are able to think, analyze their environment, and converse with their Users, offering advice or commenting on their actions. Their personalities are diverse, but most sentient Stands follow the command of their Users regardless and act to protect them.


Some other Sentient stands from Part 5 include Sex Pistols and Baby Face.  All of the individual bullets have unique personalities of their own, likewise with Baby Face.