I'm installing a gfci in a bathroom replacing a regular outlet. Initially it was tripping when the adjoining light switch was flipped. I moved that switch feed to the line side to fix that issue but my gfci tester shows ground fault error and will not trip the gfci when test is activated.
Your GFCI tester is correct with an open ground it will not trip the GFCI
the test button on the receptacle will function, this is normal for a standard 2 wire circuit.
The GFCI won’t create a ground it detects an imbalance in the current between the hot and neutral, more than 5 milli amps or 5ma imbalance will trip the GFCI’s circuitry and turn off the power.
Why is the plug in tester not working? The plug in tester places a resistor from the hot leg to the ground lug and created a imbalance. No ground or open ground and the tester can’t create the imbalance.
Is your circuit safe? It could be. This is normal for a 2 wire circuit and as safe from electrical shock as a 3 wire protected by a GFCI.
If you have a true ground that is open you should find out where ground was lost. But if a 2 wire put the “no equipment ground” sticker on the receptacle and it is code compliant even in 2021.