The term "classification" is generally used to refer to situations where the output is from a small number of options that do not have any structure, such as an inherent order (the term is sometimes still used in cases where the output does have a structure, but usually it doesn't).
The definition of "regression" is often given as meaning that the output is continuous, but there are a finite number of values that a float variable can take, yet we consider it to be "continuous". There is a point at which a large enough number of possible outputs, with a clear numerical structure of order, etc., is considered to be "regression".
For instance, if SAT score is considered to be a predictive model of future college grades, it would generally be considered to be more of a regression than a classification model, even though there are a finite number of possible scores. Netflix's percentage match is regression, even though there are only 101 different possible percentages. Something along the lines of the Homeland Security Advisory System is debatable: it has a small number of categories, but those categories are ordered. Making it even more complicated, classification models often are derived from a regression model. For instance, if you're training a model to detect "cat" versus "noncat", you likely are going to calculate a "catlike" score, and output "cat" if it exceeds a threshold.