It depends on where you look. Being inseparable certainly isn't the case globally, but it might appear that way in certain countries (or areas of those countries).
In Greece, for example, I've been told by people who live there that the punk scene isn't the only one that's anarchist-friendly. There are hip-hop shows and raves and parts of other social scenes that tend to get a lot of anarchist participation, as well as anarchist projects that don't have much to do with a single cultural identifier. Maybe "anarchist culture" or "radical culture" if anything.
On the other hand, my understanding is that certain areas have strong anarchist-punk connections because punk functioned as a way to keep anarchist ideas alive in a personal and cultural context when overtly political activity wasn't very strong or possible.
To look at the opposite side of this, it seems obvious that punk itself doesn't necessarily hold anarchism as an inherent component. There have been so many arguments about the definition of punk that it's hard to declare that there are any shared values within punk as a whole. Or even shared music tastes, a lot of the time.
In some cases, even the overlap between punk and anarchist circles is a little superficial. A lot of the classic "radical" figures in punk history are really just hardcore liberals (see: Jello Biafra, Henry Rollins) and it seems like, sometimes, the politics of punk and hardcore scenes have less to do with anarchy and more to do with veganism and animal liberation with some nods to anarchism.
I guess probably there are still a lot of connections between punk and anarchism because punk has a rebellious, anti-establishment, revolting youth sort of aesthetic that blends well with anarchism - but I don't think the two are inseparable, at all.
(That being said, even though I hate most punk music now, I did sort of come into anarchism through punk rock...)
