Hello Derek, I figured it out, so I'll share a picture of what I was seeing and why it confused me at first.
See the image below:
Let's call the services "top" and "bottom" for convenience. I've circled the "pencil" symbols. You'll notice that one of them (the top one) is hollowed out; this is a service that was shared by a student to the GIS Server rather than the portal. You can see by the single person on the righthand side (rather than the "multiple" symbol that it was not actually shared anywhere when published as the student was not connected to the portal. Now, I didn't think to click on the person symbol first, I just noticed that the pencil symbol was hollowed out (on the top service here). This service did not show up in Portal, only in manager/the rest services directory. When I did check on the bottom service I was able to get in to the editing mode as the "pencil" tool was selectable. However, the sharing settings aren't there (in the manager Editing service panel); so I thought this time of clicking on the "person" symbol. This gave me the "The service is owned by another user. The sharing properties can only be modified by the owner." issue that I noted in the first post. This led me to think that you can't adjust sharing properties at all through GIS Server unless you have an administrative account (possibly?). This led me to conclude that my only option was to delete the service and tell the student to re-publish with the correct sharing settings, or to elevate my privileges on the GIS Server (But before doing the latter, I wanted to understand more properly what was going on as disabling administrative control through the web adaptor is desirable).
However, what I didn't check at that time was that the bottom service actually *was* shared with Portal. Once I realized that, I went back to the top service and clicked on the sharing button. Once I do so, it says, "This default service has not been configured for sharing with Portal. Click "Update Sharing" to enable this." When I click "Update Sharing", the service is actually shareable with the portal, and I can edit properties. Then, after waiting some time, it gives the same "The service is owned by another user" message. But this doesn't matter as this now appears in Portal and can be edited there.
(As mentioned, directly publishing to the GIS Server rather than the Portal is something I'm discouraging because of confusion like this)
It does seem strange to me that I can't change items from Manager due to them being "owned" , but since they are now correctly listed on the Portal and changeable there, this seems somewhat of a moot point, and I'm just writing up this explanation in case someone else comes across this conversation and had some similar confusion.