Electronic – Connecting two center-tap transformers for isolation transformer

That's the correct indication for the transformer in its original configuration.

With a normal outlet, the neutral and the ground are bonded together at the service entrance, so the tester will light two lights: one between line and neutral and the other between line and ground, since the same voltage appears across both pairings.

With the tester plugged into the transformer, there's voltage between line and neutral created by the transformer secondary, but now there's no path between your isolated neutral and ground, so now that light on the tester no longer lights, indicating "open ground".

That's a big fat unscreened transformer, so there is plenty of capacitive coupling between the input and output. A high impedance meter could easily read a fraction of mains voltage between the center tap and earth. If you flip the input connections you'll likely get a somewhat different measurement.

The measure of health of the transformer is the leakage current at near maximum isolation voltage. It's not possible to measure that easily with a simple multimeter. The voltage measurement alone is not an indication of a problem, however the tingle could be. It would only take microamps to make the meter read something like 77VAC, but a perceptible tingle would reportedly take more like 1mA.

There is no problem in earthing the secondary center tap- if it blows a fuse, the transformer is faulty.