How to differentiate between images of a gravitationally lensed object The light coming from a distant galaxy or star can be distorted en route to Earth; this distortion is caused by the curvature of space when a massive object is present. My question is, how can we tell whether these doppelgangers are from the same galaxy (especially in the case of a star)?
In the case of multiple images of a background, distant object, the answer is relatively simple. You take a spectrum of the multiple images or parts of an extended lensed image and you see whether the spectrum looks the same, and in particular whether the redshift of the multiple images are the same.

Gravitational lensing affects light of all wavelengths equally, so regardless of the path taken, the spectrum should be unaltered, except that the different paths take different amounts of time to travel from the original object to us. So, if the background source, or its spectrum, are time-varying, then the lensed images could also appear different. However, distant galaxies do not change their redshift on such short timescales and so this should still be the same regardless.