This is actually more complicated than you would think. That's because Google Drive documents are really links into the cloud. So if you just copy them, you are still referring to something. And if the "owner" deletes it, you lose the document completely.

The second reason for all this brain damage is a hole in the way Google does sync to a PC, things that are in Shared to You and Shared Drives do not appear in your Backup and Sync, so you can just look at files easily, you have to have a cloud user interface to see them which is inconvenient if you have a bunch of Office Documents in them.

So with the May 2020 system on Google Drive, here is what you have to do:

1. You can't just assign any person to own your files. They have to be within your organization. What a pain. So the strategy is to do a share and then a copy and a rename.
2. From your source account, click on the folders that you want to move. It's easier, if you just make a gigantic folder and Google drive is pretty slow about this. You have to use the web interface because only there can you see the ownership in the Google Drive cloud.
3. Then you share it to your new identity.
4. Now login into the new organization. The way that this has handled has changed. It appears in your Shared with You section but not in your Drive itself. And when you say Add to Drive, it is explicit that you have created a link to the data. That is hugely helpful because before, it looked like you had become the new owner which you have not.
5. Now change the view from icon to list view and you should see the all important owned by and it will be owned by your old identity.
6. Now you select all the files (it doesn't seem to work with the shared folders which is a pain) and right click and choose Make a copy
7. This has the annoying behavior that you can say the destination, so it just drops a huge number of files the different file names of Copy of
8. But these files, you can select them all and Move them which you can't do with the other ones. Choose to move them to My Drive and this dumps them for real. You should also see they are now owned by you.
9. Finally, the painful part of renaming all these files. If you are using the Backup and Sync application, you will want to do this on your Mac. Then you can highlight and choose Rename and it is at least a little smarter.