LED-strip PWM works with Raspberry Pi, doesn't work with Attiny45
Well, after some random tinkering I found that when I ground the attiny45 in a specific place on my breadboard setup it works without a problem. Therefore I think the problems I had had nothing to do with the chip itself, just some weird problem with my breadboard/jumpercables/etc..

Sorry for not trying that earlier and thank you all for your help!