It is quite likely that Snipping Tool is also doing this. Every camera also leaves device specific signature because of its inherent silicon sensor defects.
If you want to stay anonymous, don't share images you can't verify at the byte level. Apply filtering to decrease the low bit noise that could hide cryptographic signatures. Don't trust complex container formats.
See e.g. PPM format: https://www.cs.swarthmore.edu/~soni/cs35/f13/Labs/extras/01/...
These days entire scenes can be tweaked by AI to add unimportant but identifying marks, at a level far above signal processing tricks, like moving objects in the scene. Verify from multiple sources.
If Anthropic can manipulate text to add a watermark, what’s preventing providers from doing the same to generated images? Modifying the container format can’t protect you from a signature in the image’s visual representation, unless you apply noise to that (and know it’s enough noise/the right kind of noise to counteract whatever unknown technique they’ve applied).