I want to challenge your fundamental premise:
Why is data considered to be a discrete mathematical entity rather than a continuous one?
For example, the study of Algorithms is an important subfield of Computer Science, and there are many algorithms that work with continuous data. You are probably familiar with Euclid's Algorithm for computing the greatest common divisor of two natural numbers, but did you know that Euclid also had a geometrical version of that same algorithm which computes the longest common measure of two commensurable lines? That is an example of an algorithm (and thus an object of study of computer science) over real numbers, i.e. continuous data, even though Euclid didn't think about it this way.
There are many different ways to classify algorithms, but one way that is used, is to classify them by their "continuousness":
- Digital Algorithms (discrete-event algorithms over digital data):
- the numerical variant of Euclid's algorithm
- long-hand division, multiplication, etc. as taught in school
- any computer program, λ-calculus program, Turing Machine
- Non-digital data, discrete-event algorithms (algorithms over continuous data, which however still have a notion of "step", i.e. continuous data but discrete time):
- the geometrical variant of Euclid's algorithm
- algorithms on real numbers (e.g. Gauss' Elimination Procedure)
- algorithms on continuous functions (e.g. the bisection algorithm)
- Analog Algorithms (continuous time, continuous data):
- electrical circuits
- mechanical gyroscopes
- Hybrid Algorithms (any combination of the above)
Other answers have already mentioned Real Computation in Computability Theory, another important subfield of Computer Science.
What are the drawbacks, or invariants, that are violated in structuring data as a continuous entity in $r$ dimensions?
The only real (pun very much intended) drawback is that such data cannot be represented with common digital computers. You can think about algorithms over continuous data, but you cannot run them on the standard machines we usually use to run algorithms.
That's the main reason why continuous data is not as "visible" as digital data.
However, an implementation of an analog algorithm doesn't actually need to be complicated to imagine or even to build. For example, this is an implementation of an analog algorithm:By Andrew Dressel – Own work, CC BY-SA 3.0, Link
Now, you might say "Wait, that's not a computer, that's a bicycle", but actually, you can use it as an analog computer: it computes the multiplication of a real number $r$ by a fixed rational number $q$. Turn the crankshaft $r$ times and the rear wheel will turn $q×r$ times. You can use this to scale any real number, e.g. turn the crankshaft $π$ times and the rear wheel will turn $q×π$ times; this is something you can not do with a digital computer.