The REPL (traditionally "listener") is a kind of interpreter, because it reads textual input, and hands it off for processing. There is some definition of "interpreter" which this meets.
lispm's point is that it's not a/the Lisp interpreter. An Lisp interpreter and Lisp listener are different things. The listener is more like a "command interpreter".
You can write a very poorly featured listener yourself, by literally following the REPL acronym: (loop (print (eval (read)))). You have not written a Lisp interpreter in four operators; the read and eval functions are doing that. The following is a copy and paste from an actual session:
[1]> (loop (print (eval (read))))
(+ 2 2)
4
(let ((x 9))
(* x x))
81
Some other languages are the same way. When Bash runs a script, it's not going through the interactive command line processor that is used for interactive input, which has completion, recall and editing. That command line processor isn't what parses and understands Bash syntax.
We can write a REPL at the Bash prompt also. We can write a loop which reads a line of input, evaluates it as shell syntax, and then repeats that ad infinitum until we hit Ctrl-C:
$ while true ; do read -r command ; eval "$command" ; done
echo foo
foo
for x in 1 2 3; do printf "[%s]\n" $x; done
[1]
[2]
[3]
uname -a
Linux sun-go 4.15.0-167-generic #175-Ubuntu SMP Wed Jan 5 01:55:52 UTC 2022 i686 i686 i686 GNU/Linux
^C
The shell language interpreter is in the
eval command; we have not written a shell! The "for x in 1 2 3" syntax isn't coming from our "interpreter" loop.
Lisp's eval isn't necessarily an interpreter. A Lisp implementation which has only a compiler will implement eval by compiling, like this:
;; insert expression into lambda expression.
;; compile lambda expression, resulting in a compiled function object
;; call function object, with no arguments.
(defun eval (expr)
(funcall (compile `(lambda () ,expr))))
You can write this function in any Common Lisp (just don't call it cl:eval).
Thereby you obtain a compiling eval, even if the standard cl:eval is interpretive.