It may be wrong in your model/interpretation of my words, but it's not wrong in my interpretation of my words.
In what interpretation is it consistent for 'A square is a rectangle" means "A square is a TYPE of rectangle"' and 'The word "is" implies an isomorphism' to be simultaneously true? No matter how I cut it, the latter seems to prevent the former to me.
In that case your former statement that 'The word "is" implies an isomorphism' seems to be wrong.