Like a duplicated gene, the two seem to have evolved in different directions over time.
I've read that Torah scrolls have the same kind of transcription errors "inherited" from one copy to the next so the origin of a scroll can be traced back.
What's more likely is that one psalm was copied from the other during the Torah's authorship. Just looking at the translation you can see the use of YHWH (translated as the LORD) in one psalm, and the lack of use of YHWH in the other. This is probably because one psalm was written by J, and the other was copied by E.
Also, I'm not quite sure what kind of transcription errors are permitted in Torah scrolls. They do throw out the entire scroll if there's a significant error. I can't imagine someone duplicating an entire psalm. At most, a letter might be missing from the text.
This isn't copy protection, or even watermarking to identify who's "leaking the source".
Hand-copying text is error-prone, and copies are often made from copies, so there's the chance to reconstruct a "family tree" even after knowledge of what copied from what is lost.
I've read that Torah scrolls have the same kind of transcription errors "inherited" from one copy to the next so the origin of a scroll can be traced back.