There are several things that are all called regular expressions. The answer to your question is different depending upon which thing you want to talk about.
The three relevant distinctions for this question in my opinion are as follows:
The notion of regular languages and related things like recursive enumerability.
Individual regular languages are isomorphic, (i.e. able to be losslessly transformed to and from,) to deterministic finite automata and reducing something to a regular language demonstrate results about that thing's computablity,
so I would argue it is part of computing science.
If linguistics folks find the notion or regular languages useful, however, then we can share it. Human languages are (generally?) not regular languages but something more complex, so I would be surprised if that was the case.
I think most interested people would give the same answer to whether regular languages are invented or discovered as they would about mathematics.
The particular notation for describing regular languages, in the formal literature, developed by Kleene and others.
This is clearly invented. There's no particular reason we have to have used
+ in particular to represent those ideas. If you want to assign this notation to one or more fields, I would argue it belongs to any field that publishes papers using it.
Computer programs which take as input an expression in a language and produce an automata or something resembling one and that result can then be used to perform operations on a different input including search and replace.
The languages these programs define which usually contains some form of the Kleene Star and Kleene Plus, along with several other more complicated but useful constructions, which are generally absent from formal regular language literature to my knowledge. Backreferences would be one example of this. These programs are invented, as are their particular sets of notations. They are also instances of computer engineering rather than computing science per se. Although of course they are building on at least a certain amount of computing science results, and computing science results can be stated about these programs. For example, it is possible to construct regular expressions of this sort that produce exponential running times, and subsets of those notations can be defined where that is impossible.