I've noticed that a lot of fighter planes have the pilots name(s) painted on the side. Does this mean that pilots always fly the same aircraft?

Correct, the air-force probably wouldn’t be re-painting names on aircraft after every different flight. Sometimes during sorties pilots are asserted an aircraft.

OK cool, thanks for the fast reply! That's mad that in effect, you get your own plane!

You don't get your own plane, rather, you are assigned to one plane. No ownership involved at all.

Yeah, I realise that you don't own it. But like you said you're assigned a plane.

I heard that the names are stickers.

Actually that is completely wrong (at least in the Navy)

The more senior you are, the more you will have your name painted on the side of the plane. If you notice crew chiefs and maintenance personel are also painted on the sides of planes. When my dad first checked into his squadron he didn't have his name on a single plane. By the time he checked out (as a Navy Lt.) he had his name on 3 different planes. Basically, it comes down to seniority which chooses if and how many times your name appears on aircraft. You are not assigned a specific plane at all.

Granted, this is how the US Navy does it. I have no idea how the Air Force does it but I would imagine it wouldn't be that different.