Thanks for trying the plugin, for your kind words and for this report. You have definitely found a defect (which I will fix), but the "Fatal error" is actually a side-effect, not the real problem.
I cannot reproduce this issue yet, so I'd like more information if you can help me.
First, so I understand, you are using the "edit tags" (e.g., Media/Att. Tag) screen to add new tags; is that right? You can also add tags in the quick edit area and Media/Edit Media screens for a particular attachment; does that work? Is it only tags, or do categories have the same problem?
Doe this error always happen, or only for some tag names? Can you give me a couple of the tag names you're adding?
The "Fatal error" you report happens in the function that computes the "Attachment" column on the edit tags screen, because something else has gone wrong looking up the term object for the term you're adding.
If you leave the edit tags screen and then come back to it, does the error still appear? In other words, does it only occur when you add the tag for the first time? Does it still happen after the tag has been assigned to one or more attachments?
Finally, are you using other plugins that affect or work with taxonomies? You mention Search by ID, which I'll look at, but that seems like an unlikely contributor to this issue.
I am sorry to pester you with these questions, but anything you have time to try and report would be helpful.
Thanks again for your interest and your patience.
By the way, when you say "add a search by ID", do you mean the attachment ID or the "parent ID", the ID of the parent to which the attachment is attached? It's a good suggestion.