driving licenses – As a French national (as well as Mexican), what nationality am I if entering to another EU country (say Spain)?

The two are totally unrelated. There is no requirement for French citizens to hold a French license in either France or Spain. For example, a license that a French citizen obtained while residing in another EU country is fully valid in both countries, even if you were to take up residence.

I am not familiar with the requirements for your Mexican license to be recognised in Spain but I would be extremely surprised if it was tied to your Mexican citizenship. In Spain, like in many other countries, what definitely matters is whether you are a resident or not. If you were to move to Spain, you might be required to get a local license (rather quickly in most cases, after 2 to 10 years for EU licenses).

To the extent that citizenship would be relevant, you could certainly “play the Mexican citizen card” as you put it. You are and remain a Mexican (and French) citizen. Contrary to what many people seem to believe, the passport you use at a border crossing doesn’t change that, there is no notion of being-in-the-country-as-a-X. It can work to your advantage (e.g. EU freedom of movement rights remain intact even if you don’t have a French passport) or to your disadvantage (e.g. you are not entitled to consular assistance in France, even if you enter using your Mexican passport).