I was wondering if we can wish non-muslims on their festival and I came upon this previously asked question on this platform Can we wish non-Muslims "happy holi", "happy Diwali", etc.? where the same thing is asked but here I want to refine the question a little more by asking that if we can wish a non-muslim on their festival without having a true intention behind while wishing them just for the good relations and cordial reasons.
When we say "Wishing you a happy _______", we are just wishing them happiness on the said day. We need not study the history of the celebration or whatever purpose they are celebrating it, it is their duty to study that and we need not have to worry why they are celebrating. Just like we muslims celebrate 2 eids and people of other faith wish us happiness on that day we just reciprocate it on their celebrations days.
When a (courteous) greeting is offered you, meet it with a greeting still more courteous, or (at least) of equal courtesy. Allah takes careful account of all things. Quran 4:86
For example, a non muslim who wish us on Eid ul Adha may be a vegatarian who do not believe that the sacrifice of animal is good but he may still wish us a happy Eid. So we need not have to believe in their religion to just wishing them happiness on their celebration day.
This is just my understanding from my study, use your own study or ask a learned scholar about this issue.