Driving into Germany from France

Not like US to Canada. There is no border station to stop at. You could get pulled over and spot checked, so I would have the passports handy, as well as drivers license. An IDP is always a good thing to have as it acts as a translation of you drivers license. The only issue I am aware of is that large German cities require a sticker certifying the car as low-emission. Not sure on all the details of that one.

Both countries are members of the EU, so no customs. Both countries are also members of the Schengen Area, so no passport (immigration) checks. Just a sign by the side of the road.
But, you are in a different country. Signs are in a different language and laws are different.
You need to have your passports with you, plus driving licence and car documents, but that is a basic requirement whether you are crossing the border or not.

Having to stop at a border and being restricted about where you can travel is regarded as a symbol of all that was bad about the first half of the 20th-century. Unfortunately many places outside Europe appear to be going backwards in this area.

Be warned that there are often quite punitive fees if you return a hire car in a different country from where you collected it, even if the actual distance is not very far. If you are taking the hire car back to where you got it there's no problem.

I would be surprised if you had to stop at all instead of just driving through.

On this last this trip I was the passenger, the driver was crossing the German-Polish border at Frankfurt an der Oder going to our destination... Küstrin, just crossed the Oder bridge into Poland (and later back into Germany), nothing, no stopping, no check. I did have my passport on me (out of habit), totally unnecessary, since I was not asked to present any ID when paying with an American credit card for lunch while in Poland.