I think it makes sense to think of the world of mathematics and the real world as separated in principle. However, firstly in order to make real use of mathematics, the world of mathematics and the real world have to be connected by interpretation, including addressing/checking whether the connection is appropriate. Secondly, mathematics was originally, and is mostly still now, set up in the way it is in order to be useful, i.e., having a connection to the real world in mind.
A "random variable" is a mathematical construct, and as such part of the world of mathematics and not of the real world. When handling random variables in the world of mathematics, indeed it doesn't matter whether this represents a "real random process", because such things do not exist in the world of mathematics. It is however called a "random variable" because the people who defined it had in mind to model real random processes with it.
The problem with the second question is that it is not clear whether what you call "real random process" actually exists. There are determinists around who believe that nothing is "really" random. There are also people who hold that probabilities do not model randomness in the real world, but rather the state of knowledge/uncertainty of an individual, or a scientific community as a whole. These people still use random variables when treating probability mathematically.
Even a frequentist, for whom probability models model processes existing in reality may concede that many real processes for which we use random variables are not really random, although most believe that some are (such as radioactive decay, or random sampling and randomisation of experiments if the researcher uses proper random numbers). Ultimately "objective randomness" cannot be confirmed by observation, so I'd rather say, if a model of a real process is used by a researcher, the researcher has to do two things in order to convince others that the model is useful:
(a) argue that nothing of the knowledge of the process invalidates randomness (such as convenience sampling where a proper random sample could be drawn),
(b) show that the data behave (in all relevant aspects) about as the model implies they should behave.
This will not secure that the real process is "really random", however it suggests that we can use the model in order to learn something about what goes on in the real world, e.g., make predictions, or simply statements like "data on the effect of homeopathy look like if they were generated by a random model according to which it is the same as placebo."
