Think about the traits of what we call an ideology or a theology (both are systems of ideas). They are formed of very similar characteristics and privilege specific sets of people as the arbiters of correct knowledge in any given example. They use doctrine, a common narrative and conditioning to culture approved beliefs and behaviors in a populace. The original response was to the statement that a certain day this week was not a religious holiday. I'm writing around it as much as possible because I don't want to be misconstrued.
The point that I am afraid to make is essentially that if a doctrine/code is adhered to blindly, and that code contains a deity, we call the belief "faith" and the system of ideas a "religion". Following that, an ideology is just a system of ideas and/or beliefs without the deity or with a human deity or another entity (e.g. the nation state) to fill that role. So, whether or not we call something "political", "religious" or otherwise is just a matter of applying a specific label to human belief.
I understand the distinction between the two in a practical sense. One label to discuss belief systems in gods/deities and another to discuss belief systems related to the nation state. In practice though, these lines are often blurred and say with the former king of Thailand (Rama IX), the reverence of the population looked very much like worship to me when I lived there. The behaviors exhibited were much more religious in nature than anything I would define as political, there was a definite cult of personality in the very least. It was illegal to question or critique the status quo there and apostates were punished with prison sentences, fines or trips to attitude adjustment training centers. Characteristically speaking, if something or someone is infallible or unquestionable and questioning it/them results in punishment, then an apt descriptor for the behavior of punishment is something like fervor or religiosity when referencing the cause that is being defended.
In practice, lines are blurred and there is very little to differentiate between the defensive behaviors that people or groups of people exhibit when their beliefs are questioned (whether these beliefs are religious, political or otherwise). Thus, for me, in practice, state ideologies can function as state religions and the same mechanisms are often used to promote loyalty towards them (hymn/anthem; cross/flag; bible/constitution; use of formal ceremony etc.). Thus, it if is plausible to describe defensive behaviors, as outlined above, using terms like fervor or religiosity then it makes sense to consider the possibility that traditional and state religions can exist synergistically within a society. If so, then religious holiday, national holiday or national religious holiday are all permissible as labels to describe days in the calendar that proliferate the furthering of these belief systems...