all should together make a shared base handling model for their games
That is impossible, because each physics engine needs its own kind of physics models. General facts that define physics models (wheelbase, ride height etc) are universal and interchangable, but different games need different tweaks and arbitrary parametrs to make those models "feel" as close to the real thing as possible.
If these was a universal physics engine where you would just need to put in a bunch of real life measurements about any car and it would materialize as a realistic handling model for that car, then racing game physics would be, basically, solved and all that was left was to create better graphics, and all the games would essentially use the same physics.
This is not the case, unfortunately. Phyisics engines in racing games are ugly, un-symetrical, propraetary contraptions that don't work equally well accross all spectrum of car sizes/weights etc. Thats why a super light-weight racing cart will, for example, flip over in turns or accelerate too slowly in a physics engine meant for touring cars. Or a massive 6 tonne racing truck will occasionally glitch through the asphalt or spin violently after impact.
That is why rally games have rally physics engines which are not the same as used in Formula 1 games, for example.