Why shouldn't I have one table for multiple relationships?
What does your colleague propose as the primary key for this link table? 
Primary key columns can not be NULL of course: the table above has nullable.

There isn't any natural row identifier (which is what a PK is) in the example above (a IDENTITY column is not a primary key), therefore it fails in any modelling process. Don't even think about creating tables without some model (ERD, ORM, IDEF1X, whatever)

You'd also need CHECK constraints to ensure you don't have 3 way links.

Finally, you're straying into 4th and 5th normal form territory but for the wrong reasons.

I can't find any examples on the internet: that shows how stupid this is