As Paul stated, running car tires, enables a sidecar rig to corner faster and resist loss of traction in bumpier corners, whether braking or accelerating. The rigs you do see with these setups - they are now dedicated, full time rigs - if the sidecar came off, trying to ride solo with automotive tires would be funny, if foolish :)
There are a number of different designs of front ends used on rigs.. the tried and true BMW Earles Fork design, the similar leading link design, and the newer. quite radical center-hub steering front ends. The front ends themselves are changed for the purpose of altering the front end steering geometry.. most, street-bike based, unmodified front end rigs handle like a decrepid Mack dump truck with shot power steering, and soft tires. On these rigs, the front end trail characteristics are changed from an average of 5" of trail, to often less than 1", sometimes neutral trail measurements. It makes for a very sensitive, responsive, "think about it, and you've already made the turn" ride.
The red rig uses a center-hub front end, which is much superior to my leading link front end in camber/castor issues of an automotive tire going on edge as a leading link front end is turned - a correctly designed center-hub front end has absolutely neutral camber/castor characteristics, which makes it the most precise, sensitive front end type of the two... I had the knowledge and ability to design and build a leading link front end for my bike.. a center-hub front end is way over my head in the design department..