I think there is a connection.
If I take myself as an example it just has to be true. I can be serious sometimes, but mostly I'm kind of the funny, psycho guy. You can hear that when I play I think. And as robert said, it depends on the guitar tone. If I have a strat I'd do some blues stuff, if I have a Ibanez with humbuckers and a floyd rose you will hear some metal from it. It depends what feel your gear gives you. Another thing is your mood. If I'm angry I play hardrock/metal stuff. If I want to have a good time it's more blues stuff mixed with rock. It also depends on what I play, if I go for johnny b. goode (jimi hendrix version) I go crazy on it. When I play tears in heaven my playing calms down and gets "emotional" like it should be.
So yes, character does go into your playing.
Guitars: Ibanez RGT42DX loaded with Dimarzio Air Norton (neck) and Dimarzio x2n (bridge)
MIM strat /w maple neck. new pickups soon
Amps: ENGL Screamer 50 1x12 Combo
Effects: All I use now is a dunlop wah and a big muff.