There was that whole COD WAR thing which was pretty intense at the time, and fishing quotas have been a thing in The Sun for a while, but I agree it doesn't feel like a huge part of the general identity.
I feel like people trying to build a national identity around hating Europe are really keen on fishing. Though only as far as it's useful for that end, no one actually cares about fishermen etc. That's what's so confusing to me about all this (brexit etc): it's all a very thin "papering over the cracks". As soon as someone has to make a hard choice, they renege. How anyone still supports it baffles me.