Huh. I'm from the Boston area, 50yo, avid reader, former English teacher, fairly well-travelled...
I've definitely been using phrasing like it ("save off a copy", etc) for at least the past 20-25 years (upstate NY, moderately online, avid reader, parent is an English professor, fairly well-traveled).
Conceptually, I think that the "off" serves the purpose of aligning it with something like "split off"—you're essentially forking the history by creating a separate saved copy.
1. https://answers.microsoft.com/en-us/msoffice/forum/all/savin...
2. https://community.fabric.microsoft.com/t5/Service/End-User-S...
In both cases it's being used for "copying".
We can also see it was used as early as 15 years ago on this very site ( https://www.hackerneue.com/item?id=1182478 ), so it's not a new turn of phrase.
My experience has definitely been that I've heard it more online and in San Francisco, and not very often in Germany, Texas, or Russia. Are you in one of those areas?
I feel like the etymology is something like "print off a few sheets" becoming "copy off a sheet with the copier", and then to the more general digital copy meaning.