An ideology can be seen as a collection of rituals in action, reasoning, and rhetoric situated among core beliefs about what life is, what the world is, and what's important to them.
And each ideology has a specific history of events, founders, and later elaborators that shape what these rituals and beliefs are.
Many ideologies reject deities or are at least ambivalent about them, but they're not really operationally different than religions.
So for as much as little bits of history linger in relgious rituals, both blurred and resharpened in later years, the same is absolutely true for ideologies.
The capitalism or feminism or humanism or atheism or whatever else you point to today is similar to the one sharing its name in the past, or in some other region, but is not the same, and these differences are all vestiges of little seed events that happened here or there.
I certainly would (not as a comprehensive explanation though of course). The range of forms of conceptualization any one individual is capable of (or not) is strongly influenced by the ideology to which they subscribe (or are captured by).
If you simply consider how the human mind works, this "should" be fairly obvious. But one's ability or likelihood to think about such things is once again a function of the norms and accepted practices of one's ideology. That which is not known of, essentially does not exist.
Do you have an example? It sounds like a complete category error. That or you're using the word very differently than most people do..?
An adherent of an ideology might, like, have rituals. But the ideology isn't the rituals.
Here are some ideologies off the top of my head: democracy, marxism, environmentalism, libertarianism, atheism, the moral basis of Christianity but not the religion itself, utitarianism, humanism, empiricism.
None of those have, as far as I can tell, a single ritual inherently associated with them.
If you categorically define an ideology to be a system of pure concepts, independent of any practice of thought, speech, or action by a professed adherent, then you will inevitably see it as a category error, yes.
(You'll also have a hard time enumerating those concepts in a complete and consistent way)
But if you're even a wee bit of a subjectivist, as many (not all) social scientists and social philosphers are, then a definition like that isn't interesting or productive. From that perspective, ideologies are something that people profess adherence to and express statements about and behave in self-identified accordance to.
They gather in elections, they discuss power through the lens of capital and labor, they recycle or avoid eating meat, they recoil at government overreach, they expose fraud in purported miralces, they pray, they trade trolley problem memes unironically, they protest against inequality, etc
If these don't make sense to you as "rituals of ideologies" for people doing cross-cultural studies, that's fine, but then I have a sense that a lot of cross-cultural studies just feels like hogwash to you anyway. I doubt I could change your mind here. :)