Off the top of my head, the International Residential Code says the hole diameter cannot exceed d/3 and the edge of the hole most be at least 2" from the top and bottom of the joist. Based on your sketch it looks like you're cutting a slot, I would say no. However, could you add elbows to eliminate the slot?

I am not sure how much that piece counts in that exact position as it is at the end of the joists, toward the exterior wall in the basement
I think it is meant to keep together the double joists that are on each side of the pipes and there is already a double piece like that before
It is attached using hangers ...here is a detail
I could probably take it out, rotate that elbow in position under the upstairs toilet so that it connects back to the position of the new drain pipe
and then put the piece of wood and the stringers back

You should put it back. Take a look at the attached publication. There is some guidance about floor framing for tubs and openings.

• https://files.engineering.com/getfile.aspx?folder=58581617-bb8d-4be5-afe0-0

thanks for the link
The author of this video has a couple of videos that deal with this issue
https://www.youtube.com/watch?v=73_vpD-lLyY
also some details here
https://www.youtube.com/watch?v=74pk6faF58A

If it were my house I would have zero issues with removing it. You may need to add some other framing back in to make sure the subfloor is supported adequately, but those would not need to be full depth members.

I just realized that it is there for the only purpose that it supports the toilet. it reinforces the area under it which might cave in if not supported from underneath
real life picture below
I will slightly relocate it