I can't think of anything we HAVE to carry in the car in the UK except for a paper disk on the windscreen showing that tax has been paid (to be done away with shortly, it is all online now).

No ID, no documentation, no first aid, no warning triangles, etc. etc.

In France you need, ID, a warning triangle, two breathalysers, safety vest in the car, not in the boot (trunk). Belgium also needs a fire extinguisher.

In Germany it is illegal to run out of fuel on the autobahns and winter tyres in the winter.