I don't think it's been as universal with either fanbases within the city but there's always been an air of indifference from a decent amount of both sets of fans peppered with an active dislike for the national side. I'd say it was probably stronger with Evertonians for a decent period cos kopites were latching on to the Owen-McManaman-Gerrard-Heskey etc. presence in the side.

It's only more recently as that's waned and the whole identity crisis their local fans went through and how they needed to reinforce a scouse identity and "reclaim the Kop" and "keep flags Scouse" lest the sense that Everton are the club for scousers and Liverpool the team for tourists fully set in. The more they've leaned into that "Scouse not English" mentality the more there seems to be a reaction from a number of blues who seem to think having even a level of ambivalence towards the national side is "kopite behaviour" which I just feel is a bit sad really. Be as interested in England or international football in general as you want, it's really not important, just seems a bit sad when people do it just based on what fans of other clubs think or do.