Inherently discrete variables (like "count data") have special properties, and it matters when it comes to econometric estimation.
The usual criterion to treat a variable as continuous or discrete, is I believe not what we observe but what could be conceivably observable.
Example: if one counts number of people, the variable is inherently discrete. In large samples, the continuous approximation may not matter -in numerical analysis, starting the search for a solution by "pretending" that the data is continuous is done all the time. But in small samples it may matter. Also, there are variables that take only a few specific values (usually integers). Here, taking into account the discreteness is crucial.
Prices have no reason to be treated as discrete. They are continuous because in principle nothing constrains a price to not take any value on the half-real line.
As for the data sets: data is always discrete -simply because between any two real numbers there is an "infinite" number of other real numbers (with the appropriate mathematical definition of infinity). So since the sample is always of finite size, it will always be discrete. The issue is whether the Data Generating Mechanism can be conceived as being continuous or not.
Finally, for certain purposes continuity is a great analytical helper bar one-dimension: the time-dimension, especially in stochastic setups. Continuous stochastic calculus notoriously carries a mathematical baggage that many times (but not always) one feels that is not worth the trouble.