This behaviour comes from the
TeX-insert-dollar function which is described as
(defun TeX-insert-dollar (&optional arg)
"Insert dollar sign.
If current math mode was not entered with a dollar, refuse to
insert one. Show matching dollar sign if this dollar sign ends
the TeX math mode and `blink-matching-paren' is non-nil.
When outside math mode, the behavior is controlled by the variable
With raw \\[universal-argument] prefix, insert exactly one dollar
sign. With optional ARG, insert that many dollar signs."
Looking at the implementation there does not seem to be a possibility to only disable this specific behaviour of refusing to insert
$. You basically have two options:
Follow the documentation shown above and use the universal argument, i.e. press C-u $ to insert a dollar sign.
Remove the $ key from the mode-dependent mapping, i.e.
(define-key TeX-mode-map "$" nil)
This will map $ back to
self-insert-command (or whatever is in your global map) but this way you will also lose the other functionality brought by
TeX-insert-dollar such as