My current guess is that you get much or most of the benefits, but not all (by both value and number). If you look at the actual changes in the body during both of these activities, most are the same as exercise, but not all.

For example: body temp increases, heart rate increases, and we sweat. But the muscles aren't "engaged", consuming stuff (glycogen, etc.) while doing sauna.

There could also be sauna benefits that exercise does not impart, or is less likely to do so: sweating greater than exercise could lead to excess excretion of plastics, carcinogens, etc.

Running in mild/cold temps we do little sweating (unless long duration exercise), whereas every darn sauna at sufficiently high temps we are going to be sweating.