I think what you’re trying to do is reasonable but Airtable cannot group by linked records (yet).
The only 2 options you can use for grouping are the “single select field” or the “collaborator field”:
As a hack you can create a new single select field specifically for the kanban grouping. Something like:
Then selecting all the project cells and paste them into the new field. Before the copy:
After the paste:
The new field is now selectable when setting up your kanban view:
You’ll now see the grouping you’re after: