We didn't really have a better word for units of temperature, as "Celsiuses" or "Farhenheits" didn't become a thing, so "degrees (of) Celsius" or "degrees (of) Fahrenheit" is kind of what we're left with. We could have chosen to say "degrees (of) kelvin", but we decided to go with "kelvins" (like "meters" or "seconds") instead. It's just a choice of language.
Temperature does have dimensionality, but it's weird because it's a bulk property of a particular material. If you consider two stars, you can add their masses, or their volumes, and the resulting values are meaningful. But you can't add their temperatures. You can't even average their temperatures meaningfully without knowing the relative sizes of the stars and a bunch of other information either.
That said, I can't think of a situation where degrees (of temperature?) would disappear in a mathematical operation. Can you give a specific example?