Setting verbosity level from list of function arguments

I'll start by saying that a blind scan of the arguments like that would fail to find all the valid ways the verbosity is to be specified if we assume the aim is to accept GNU-style long options.

What if the user passes -xyv3 (same as -x -y -v 3) or --output -v2 (same as --output=-v2, where -v2 is the argument to the --output option) or cmd -x -- foo bar -v baz (where -v is not to be treated as an option), or --verb=12 (abbreviation as supported in GNU option style parsing)? getopt / getopts are there to help you parse options properly.

But if you don't care about proper option parsing, and can rely on the user always passing the -v / --verbosity option like that and not passing those strings as non-option arguments otherwise, you could do:

extract_verbosity() {
  unset -v REPLY
  while [ "$#" -gt 0 ]; do
    case "$1" in
      (-v)
        [ "$#" -gt 1 ] || break
        shift; REPLY=$1
      ;;
         
      (--verbosity=*)
        REPLY=${1#*=}
    esac
    shift
  done
  [ -n "${REPLY+set}" ]
}

and then in your function:

my_function() {
  verbosity=0 # default
  extract_verbosity "$@" && verbosity=$REPLY
  ...
}