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I am french and attended a private university in France for my college education. I currently have a Masters in Computer Science from this university and also worked there for two years as an adjunct lecturer after my degree. I have an additional 5 years of experience in the Software Engineering industry.

Said university is called "Catholic University of {City}" and I fear that it may convey the wrong idea about the education I received and the kind of person I am if I put it as such on my resume.

Many private schools (from elementary school to college) in France are "catholic schools" though the education they provide is secular and they are attended by students from all range of beliefs.
In the past, classes in such schools may have been dispensed by nuns or vicars, but nowadays they only employ regular teachers and lecturers.

My university is under contract with the state (so are most of french private schools) and the content of the classes I attended for my degree were scrutinized and "vouched for" by a public university. The diploma I received bears mention of this public university though I never actually attended any classes there.

The university is well known in France and it will be understood by a french recruiter that graduating from a catholic school does not entail that I received a religious-oriented education or that I am myself a catholic/religious person.

But how should I put this on my resume if I am applying abroad to avoid such asumptions being made ?

Should I mark it down as being a graduate of "Public University of Y" seeing that they are the university mentioned on my degree even though I did not attend there ? Or am I just making a big deal out of this and people will most likely not care ?

It is a non-issue. In the US there is a major university in Washington DC called The Catholic University of America. I have known many graduates of the undergraduate, graduate, and Law programs of Catholic University. A student doesn't have to be a Catholic in order to attend. While there is a requirement for a small number of religion classes they don't have to be about being a Catholic.

There is always an issue when the University is not in the same country as the person reading the CV/Resume. The reviewer is most likely unfamiliar with all the universities in your country, but a quick google search should clear up any issues. I wouldn't try to disguise the Catholic part of the name, it would just make it harder to find the correct school to determine the quality of the University.

I graduated with a Bachelor's Degreee in Computer Science from a fairly well-known, very conservative Christian university in the United States. When I was looking for a job, I was concerned that potential employers would see what school I went to on my resume and reject me for not being a good fit for their company culture. (I am a conservative Christian myself, but I'm tolerant and respectful of others with different beliefs.) I decided to put my school's name in my resume anyway.

The only question I received in interviews about my school was why I went so far away from where I lived to get my university education. I replied, "My school had the best Computer Science program of all the Christian schools I knew of."

I ended up getting an excellent job at a company where I would say that pretty much everybody else has radically different views than I do, but it isn't a problem for them or for me because everyone is professional and respectful of each other.

Based on this experience, I would suggest that if an employer decides to overlook you for your school's religious affiliation (which I doubt would happen), you probably wouldn't be happy at that job anyway. Thankfully, there are places (hopefully most places) where people are professional and don't care what you do or don't believe as long as you have good qualifications and can deliver on what you say you can do.

Don't worry about the name or even if your University was a religious school. It's much more important to be honest and clear about your education than to be vague about it and raise concerns that you might be ashamed of your degree or think it's somehow inadequate. It's not.

Good luck on your job search!