GFCI tester doesn't trip GFCI. Bootleg ground?
It is normal for a plug-in GFCI tester to not work, if the receptacle is not grounded.   GFCI testers are not magic.  

GFCIs work by comparing current on the "hot" wire to current returning on the "neutral" wire. If all is well, they should be equal.  if they are not equal, some current is traveling down a third path possibly through a human.  

If a third path does not exist, then tripping a GFCI is impossible.   An external tester cannot trip a GFCI if it only has access to hot and neutral.   The GFCI itself can test itself by using an internal path which goes around the LINE and LOAD terminals, both of which it has access to.  An external tester can't do that.  

So failure to trip at a downstream outlet does not tell you anything one way or the other about whether ground has been bootlegged there.   It only tells you ground is not present.   Such a receptacle should have "GFCI protected" and "No Equipment Ground" stickers and will need those to pass home inspection. 

To use an external tester at such a location, get a common 3 to 2 prong cheater, the kind that has a 3 inch green pigtail meant to be attached to the receptacle screw.  Extend that using whatever wire you have on hand, all the way to the panel, attach it to the building's grounding electrode system.   That becomes the third path.  

You are allowed to retrofit proper grounds to a receptacle without disturbing the hot/neutral wiring, and you don't even need to use the same route.   (Also remember metal conduit is a ground path).  However if a circuit has GFCI protection upstream, I would call such an upgrade a low priority.  In some ways, GFCI protects better than actual grounding.