That's political disagreements though not ideological disagreements. I think if you look at the ideological thinking on either side they're pretty heterodox and fluid. To be honest I have a hard time keeping up with where the GC side is at now. These are not centralised movements with doctrinaire lines. When people start talking about ideologies, it gets my heckles up because it's one step away from outright conspiracy theorising. Do these ideologies not have a real social basis? And I think it's common for people to rally to certain causes together while in ideological disagreement - trans people are in it together not because they share an ideology about gender, but because they are dealing with the same threats to living peacefully in society without battery and with access to trans healthcare.
There is a political ground that's popular in the US and the UK (at least) that sits between that hazy space between conservativism and fascism. Think Nigel Farage here and Donald Trump there as political figureheads. This milieu is all about free speech and the agenda of the "liberal elites" and their "woke ideology" that wants to suppress them ie. the "ordinary people" (ie. white/cis/straight/male people who at best are ignorant and uncomfortable with challenges to their thinking and at worst outright bigots or even organised fascists). I see it all the time both irl and on line. They were galvanised in opposition to feminism some years back and then (in the US in particular), they were galvanised by the Black Lives Matter protests. They've moved on to trans people and environmental activists now and who knows what next. They see their way of life under existential threat. Sometimes the threat is explicitly stated in terms of "cultural Marxism". And no I don't see such people as following an explicit ideology because it's way too messy to be identified as something so definite, but they do characteristically see things in terms of concrete ideological enemies.
This is why I don't think it's good to think in terms of ideology and the clash of ideologies, because in doing so you're dipping a toe into this sort of thinking.