Transformers can be tested, yes.
Quite possible that it's a custom part, in which case you'd have to ask the manufacturer or customer for a drawing. Good luck with that...
The first steps are:
- Noting dimensions (height, ID, OD)
- Wire size if visible (should be able to guess at the outer layer, anyway, and figure out which color wires it's on)
- Number of turns if visible (pretty much N/A here)
- Measure DC resistance of each winding, including mapping out which colors are connected to each other
Then you can guess at primary, secondary, CT or multitap windings, etc.
Pretty safe to apply mains frequency, at low voltage (say from a 6.3V transformer?), to most any winding, and measure the other voltages. This gets ratios.
Ideally, you'd do a saturation test as well, using a current-limited source (maybe a transformer with a series resistor) and an AC voltmeter. Apply relatively high voltage to a probably-low voltage winding. See where it starts dropping off: that is, as you increase ACV, the winding V will rise proportionally, until it starts to shy away (rising slower and slow) at some point. That point is the onset of saturation, and is (more or less) the highest voltage that is reasonable to apply, at that frequency.
Finally, current rating of each winding can be guessed from overall size, and winding resistances.
Tim