I managed to resolve the issue.
In my case, the problem was caused by a plugin that automatically updated Livewire to version 3.5.16. This update introduced the issue with the modal backdrop.
To fix it, I forced Livewire to revert to version 3.5.12 by editing my composer.json file and
"livewire/livewire": "3.5.12"
Then, I ran the following command to apply the change:
composer update livewire/livewire
After downgrading Livewire, the issue was resolved, and everything works as expected again.