What about "$(dirname "$0")" ? The chance that the directory doesn't exist is null, and if cd fails for any reason you get the wrong path (you can use set -e for this).
The same applies in various places in the script.

You can make it simpler with:
if ! git fetch "${GIT_REPO}" tag "${TAG}"; then
or:
git fetch "${GIT_REPO}" tag "${TAG}" || (echo echo "Error: Remote does not have tag '${TAG}'." && exit 20)
It has the advantage to work with set -e as well.

This is repeated several times in the script.

echo is unlikely to fail, but it is generally safer to use set -o pipefail

What is the reason for adding the assets found rather than asserting they are all present ? Is there a case where you want to create a draft release with only some of the assets ?
Furthermore the asset dir is mandatory, but allowed to be empty which is confusing.

Check for jq (it's not a bash built-in nor a very common command AFAIK): if ! command -v jq; then echo error; exit 42; fi

See previous comment, is that an expected behavior ?