TL;DR That is 100% correct functionality.
A magic 8-ball tester is really just 3 lights. One between hot & neutral (expected to be on for a good circuit), one between hot & ground (expected to be on for a good circuit) and one between neutral & ground (expected to be off for a good circuit).
Since you don't have a ground wire connected, there is no current flowing between hot & ground, so that light is off which is interpreted (correctly) as "open ground".
When you press the GFCI test button, what you are actually doing is sending some current from hot to ground. That is normally not a good idea, which is why we have GFCI in the first place! If everything were connected properly, then that would result in the GFCI tripping. In that situation, if the GFCI is functioning properly, power to hot would be stopped. That would result in no lights on very quickly. So the lights while pressing the GFCI test button are undefined and meaningless - and normally not visible.
Since you don't have ground connected, while you are pressing the button you have:
- Hot to neutral - since the switch has no reason to interrupt this.
- Hot to ground - since internal to the tester hot is connected to ground (deliberately) so the light showing hot to ground turns on.
This is equivalent to a hot/neutral reverse. In a hot/neutral reverse, the hot to neutral light is still on (120V between them, just "opposite" but with AC that makes no difference) and neutral to ground is on because neutral and ground are 120V apart. In this case, hot is being sent to ground, so that makes ground 120V apart from neutral. The end result is that, technically speaking, when you press that button you are simulating a hot/neutral reverse, even though what you are really trying to do is create a low-level ground fault.