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”:
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As a hack you can create a new single select field specifically for the kanban grouping. Something like:
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Then selecting all the project cells and paste them into the new field. Before the copy:
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After the paste:
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The new field is now selectable when setting up your kanban view:
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You’ll now see the grouping you’re after:
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