Different units for the same quantity usually only differ by a multiplicative constant, such as inch and meter, but °F and K differ by both a multiplicative constant and an additive one. Does that make them so different they're really measuring different quantities?
If they're the same quantity, then where's the limit of what operations you can do before it becomes something different? Is 1/s still a unit of time or is m²/s² a unit of velocity? In these cases, instead of multiplying and adding a constant, we're inverting the value or squaring it.
I've heard them distinguished as "thermodynamic temperature" and "temperature" in SI but that seems to be uncommon and it's hard to find justification for it.