crossing Schengen borders by car

No.

But you need to check the fine print. Rental car companies limit the number of countries you can bring the car to. Germany to France should not be a problem, but better safe than sorry.

Although for a city visit, consider taking the train instead.

Just be upfront with your rental car company that you are venturing into France. Policies vary by company. I know that Hertz required us to disclose going from Italy to France. There was a nominal fee. As Badger noted, better safe than sorry.

We also were charged a nominal fee when going from Croatia to Slovenia.

but so far as the actual crossing, many crossings have an area for trucks to stop for possible inspection and the road widens where the car checks used to be. Other than that there will often be a short reduced speed area which most people make a token lift off the gas, and there will be signs indicating the next country (like going into a different US state or Canadian Province) with the National speed limits. The road surface often changes, just like going from state to state. Other than that, usually no issue.

Switzerland and Austrian often have a check, but that is for vignettes and for Swiss cars, Customs.

Freiburg to Colmar, the bridge has a McDonalds on the German side of the border, and slow across the locks on the river, Strasbourg could be a ferry (free) or country bridge or highway bridge. No issues.

But DO notify the rental company.

If you are going to different countries, shop around for car hire. Some companies just charge extra for the actual day(s) you are in the second country, whereas others will charge extra for the whole hire period. Some may not charge extra, although most do in my experience.

Although there are no physical borders, taking it into a second country without the hire company’s agreement will invalidate the insurance.

I suggest you go online and read the Terms and Conditions of the rental car company you anticipate using. I believe Sixt can hit you up for $ when crossing borders. I also don't like some of their other terms regarding insurance.

Some U.S. car rental companies are franchised in Europe and they don't necessarily work the same (or as well) as the same name does "back home." I usually stick with Hertz or go through AutoEurope.