How much can a 700x23 inner tube stretch?
Yes the tube size should match the tire size.   But a smaller tube will stretch.   Too big a tube may fold.

Max means max.  If the the tire says 85 psi max then that is the max. 
If you weigh 180 then run the tire at or near the maximum.

Do not put 116 PSI in that tire.  Even if you have the wrong size tube pressure is still based on tire (not tube).