As well it should.
Normally, loads are connected between hot and neutral. Appliances are not supposed to connect hot or neutral to ground; ground is only a shield.
The GFCI compares current on the hot and neutral wires. They should be the same. If they are not, current has found another route, possibly through the grounding system (which isn't supposed to happen) and potentially through some poor human.
Circuit testers are trying to test whether ground is connected... cheaply. They mis-use "hot" as a power source, by connecting a light bulb between hot and ground. If ground is connected correctly, this will light.
In other words, it intentionally creates a hot-ground fault (by sticking a light bulb there). This is exactly the condition GFCIs are designed to detect.
I'm not talking about any ground-fault-test the tester may also have.
So why do testers often work? With a perfect GFCI, they wouldn't work. I suspect it is because GFCI's have detection thresholds above zero, and that is often enough for these testers to "get away with it". I even think there may be a tacit agreement among manufacturers for this, but obviously, lower sensitivity impinges safety. Remember a shock which only stuns you can kill you with secondary effects like falling or drowning.
So either your GFCI is pretty good, or your tester is pretty bad.
Never never NEVER connect the ground of an outlet to neutral!!! Depending on the situation, you can end up with a dangerous or even lethal voltage on the ground and therefore, on the case of something plugged in.
It is legal and advised to use a GFCI to install 3-prong outlets on an ungrounded circuit. This also applies to standard outlets downstream of a GFCI. The outlets must be labeled "GFCI protected, no ground" (labels should have been provided along with the GFCI).
You should remove all of the "bootleg" grounds.
BTW, the reason the tester didn't trip the GFCI is that it works by connecting line to ground through a resistor to simulate current leaking to ground. However, because the outlet ground is actually connected to neutral, there is no actual leakage.
Best Answer
If you were to read this: HOW GROUND FAULT CIRCUIT INTERRUPTERS WORK.
You can see that all you need to do is utilize a 15K Ohm Resistor across the Line and the Neutral to trip it. Pages 14 - 18 of that document should reflect this. You can make your own tester this way.
As noted by a commenter the resistor needs to be placed from neutral prior to the sensing circuit of the GFCI to the Hot After the GFCI sensing circuit. This will appropriately create a TEST circuit like that which is found in the documentation linked to..
While there are ways of getting around the need to remove the outlet from it's box it should not be attempted by anyone who does not know what they are doing. I will leave that to those who are knowledgeable enough to figure it out.