Car travel from France to Spain to Portugal

No requirement to change cars. Although if you're hiring the car you need to check any restrictions there are, including potentially a charge for picking up in one country and leaving in another.

No need to change rented cars at borders. No restrictions on taking rented cars across borders, in western Europe at least. All there will be at the border is a sign, giving the national speed limits. The old border posts have all been boarded up or demolished.

But, you will need to return the car in the same country you picked it up (France), or you will nave to pay a very large surcharge for someone to transport it back.

Make sure you learn the road signs and traffic laws for each country. Road signs will be mostly the same, but may be different from what you have where you live.
All 3 countries charge tolls on their autoroutes/autopistas/Autoestrada
If your licence is from a non-European Country you will need to get an International Driving Permit (IDP) to certify your licence. You need to get in the country that issued your licence.

Curious......
How many days is your journey?
Have you calculated the total amount of time you will spend driving?
How many people with how much luggage will need to fit into the vehicle?
When going from point A to B and as part of the journey and you stop for a few hours, "What will you do to secure your luggage from theft?"
What time of the year shall you travel?
Have you previously done a similar trip?

You might want to go on GoogleMaps and track your proposed odyssey.
Spain and France are much larger countries than many travelers realize. And budget airlines like Vueling make it fast and efficient to go long distances there.
Return fees can be substantial crossing country borders. Mixing up flying and driving can be beneficial as you can get the best of both worlds.

Thank you all for your excellent responses. We are in the preliminary stages of planning this trip for early October 2019. We are considering flying to Biarritz, driving to San Sebastián and south to Lisbon. Flying back to the US from Lisbon. Or, maybe flying to Barcelona, train to San Sebastián, eliminating France altogether. Still in the very early planning stages. Will take all your advice into consideration. Thanks.

To save 1000€ in drop off fees, we turned in our car that we rented in Lisbon, took a taxi across the Portugal -Spain border northof Porto and picked up a Spanish car rental in Galicia. Our trip went from Lisbon to Barcelona, six weeks total. We were in France for part of the trip but returned the car we rented in Vigo, Spain at the Barcelona airport.

Consider flying R/T Spain (either Barcelona/Madrid, or Madrid/Madrid). Pick up the car and loop north through the corner of France and Galicia, Basque country, Portugal and drop the car in Sevilla (well worth a couple of days). Then train to Madrid. Or skip Andalucia and go straight to Madrid.

Make sure to let the car rental agency know where you intend to take the car. We had a company in France that would not let us take the car into Slovenia. So, we had to get a different car company in Germany that did allow us to drive the car there. We normally rent cars and drive them into multiple countries and that was the first time we encountered this problem.