Just to add to Anna's information.
Each country offers it's residents the opportunity to document their vaccination, test, or covid recovery information on a digital app, that provides a QR code that indicates to a business or venue whether you are "safe" to be in confined areas, without disclosing private health information. The same can be done by having a printed QR code, so digital is not required.
Within the EU, there is an agreement among the member countries to accept vaccination status from another EU country, and measures were taken to make the various digital systems and QR codes compatible, so a French app or code, will work in Germany.
However, the various countries vary on whether they allow non-residents to obtain a QR code, France does, Italy does not. But most countries do accept the US CDC card as proof, but then again, some, like the Netherlands, do not.
There may be independent apps out there, but a word of warning, the EU countries do not accept, or better stated, can not scan a QR code that is incompatible, like from a third party app, or even the Washington State code, so you either get an EU countries code, or use your CDC card.
Since you are interested in France and Spain, France will issue their Pass Sanitaire to visitors, and you can apply for it online...the last I knew...and have it before you go. If you look through the "France" discussion board, you will find a wealth of information.