How to measure the maximum current a transformer can handle?
There are no hard answers to this question for mere peasants like us, but there are some rules of thumb.
As has been stated, measure the size of the core and compare it to catalog listings & drawings. This will give you an estimate of the transformer's total power capability.
But if the transformer has multiple secondary windings, you still have to figure out the output power for each winding. One reference that I saw said to measure the open circuit voltage and then load the winding until the voltage drops by 10%. Use that as the load rating. I would then total the numbers for all the secondary windings and compare it to the rating based on the core size. If the winding total exceeds the core size value, start derating the windings until everything lines up.
This only applies to mains frequency transformers. If you're dealing with a transformer for a switching power supply, all bets are off.
Note that if you're trying to measure temperature rise, it could take an hour or more for the temperature to reach steady-state and, even then, the temperature in the center of the transformer probably can't be measured and will be much higher than the surface. Also, it's normal for a fully-loaded transformer to be too hot to hold onto. But, cooler is better for longevity of both the transformer and the other components near it.