This scale is then treated as continuous, despite the possibility that differences between each interval may not be equal.

I don't know exactly what you mean by this. Can you elaborate?

I can't give you a more in-depth answer to your question, but I hope this puts you on the right path:

This is often called scoring or assigning a score to an individual. This is very common in epidemiology. I usually see it from 0 to 100. It's an easy metric to assign to people in your study. I know this is how nutrition is almost always assessed. You can look up the Healthy Eating Index for an example of this.

To answer you question with another question: If you don't treat interval data as continuous, what would you do instead?

It's not the best practice to treat an ordinal variable as continuous, but it's certainly not uncommon. A ordinal variable with 13 levels eats up 13 degrees of freedom, and it's hard to interpret. The wider your interval is the worse this problem gets. Sure, the interpretation isn't exactly right if you treat an ordinal variable as continious (what is an 8.3 on the depression scale, for example), but it's pretty close.