Axiom's user base are mathematicians in some domains. The target audience is just not web developers, Linux sysadmins, IT consultants, or whatever profession there is.

Axiom went through a period of research, but was commercially released and maintained for several years. It is now around 40 years old.

I would agree that Lisp is not that suitable to industry in general.

Lisp has been developed to do symbolic computation. Computer algebra systems fall into this category. These have applications in physics and many other domains. You would need to scan the large bibliography of some of these systems to find all the application domains. Remember Mathematica is commercially sold and widely used. Axiom fall into this category.

Lisp can be naturally used in all the domains where one computes with all kinds of symbolic data and languages. Another typical example are microprocessors. Lisp has been applied to check the correctness of various operations (floating point, ...) of AMD processors. This is not 'research', but part of the testing of chip designs. The chip design will be described in some language and some software will prove that the operations are correct based on a set of requirements.

These are natural Lisp applications and they have nothing to do with research, prototyping or templating. They are applications of computing with symbolic expressions.

This means Lisp has a natural application domain. Though this domain may not be large or popular, it does exist and provides a reason why it still exists.

McCarthy wanted a language for certain problems. With lists, functions, symbolic expressions, evaluator, ... and more. Then he and the people he was working with stumbled across the thing that it can be applied to itself. This is why we say Lisp was not invented, but discovered. Lisp was supposed to have some kind of conventional syntax. But it was discovered that one can output the internal data that represents data and code as s-expressions. Then one saw that those can be written and read back. So the internal symbolic data suddenly had a corresponding external data representation. One that could be used for data, all kinds of data. Suddenly also Lisp programs themselves were trivially externalized data via s-expressions. Read back they could be manipulated with the usual operations that were designed for symbolic expressions.

One application of Lisp's capability to compute with symbols, is that it is applied to itself and thus provides meta-linguistic capabilities (macros are well known). This means that just as easy as you can write language for symbolic computation with mathematical formulas (for integration, differentiation, simplification, and all that stuff), just as easy one can write and extend an implementation of Lisp, since its programs are also symbolic expressions. So the capability (computation with symbolic formulas) that was developed for domains like maths, planning, scheduling, knowledge representation, theorem provers, ... can be applied to Lisp itself.

This has interesting effects and is also the reason why Lisp is especially disliked for tasks where flexibility is not needed or not wanted and programmers use a language without meta-linguistic capabilities. Lisp would require more self-discipline and in industrial projects fixed processes with fixed tools are often preferred.