Grep all lines with exactly one instance of a specific character

try

grep '^[^#]*#[^#]*$' file

where

^      ; begin of line
[^#]*  ; any number of char ≠ #
#      ; #
[^#]*  ; any number of char ≠ #
$      ; end of line

as sugested, you can grep on the whole line, with

grep -x '[^#]*#[^#]*'

with

• same pattern without begin of line/end of line anchor.
• -x to grep whole line, see man grep

-x, --line-regexp
   Select  only  those  matches  that  exactly  match  the  whole line.  For a regular
   expression pattern, this is like parenthesizing the pattern and then surrounding it
   with ^ and $.

Using awk:

awk -F'#' 'NF==2' infile

based on # field separator, if number of fields in a line was exactly two fields then will print out. note that for example #x or x# or even # are considered two fields so.

With two calls to grep: pick any line that has at least one #, then remove those that have at least two:

grep '#' filename | grep -v '#.*#'