Airtable is working as designed. The “allow linking to multiple records” is a user interface setting preference only and this choice is not enforced anywhere else in Airtable. 

I’m not saying that I agree with this design choice. I’m saying it is a long established fact. 

A common workaround to avoid an item being in multiple boxes is to create a filtered view that only shows items not in boxes. Then in the linked record configuration in the boxes table, limit selection of records to that filtered view. 

Note that it will still be possible to create a many-to-many situation by setting linked record values on other ways, such as copy/paste, scripting, and the web API.

As for moving an item from one box to another, that is a different matter. You could group your items by box and drag your items from group to group. Note that this is done from the items table and not the boxes table. 

You could use an automation that watches for changes in an item’s box and when an item has two boxes, the automation removes it from the previous box. But automations are slow.

Thank you for your response, kuovonne and for providing tips how to work around this issue.

I found this blog post from Airtable: https://blog.airtable.com/how-to-establish-relationships-and-rules-within-a-database/

It says:

"For example, if you specifically define a relationship as one-to-many, you can enforce a rule that the records on the “many” side of the relationship can, indeed, only ever be linked to one record in the other table."

With what I'm observing and what you're confirming, this is not the case then, right? As in, I can define a relationship as one-to-many and still link records on the "many" side to multiple recodes in the other table.

Is there a way to file a but report with Airtable about this? It's a strange oversight that you can't more items from the view on the many side.