Hi,
it looks like you have WordPress 5.2.3, I tried on recent WordPress and the plugin worked there, the plugin uses the standard approach to saving settings in WordPress and is working for me, so likely something in your environment.
If you open up the developer tools (Ctrl + Shift + i in Chrome), pick the “Network” tab, and then load the settings page, when you save settings you should see an “options.php” request added to the network list, you should see a “302 Found” in the source view of the Response Header, and you should see your settings in parsed “Form Data” section.
Should look something like this:
https://waters.me/WordPress-options.png
Take a look at that, and check the “error log” from your WordPress host for entries that appear when you submit the settings you want. If it is getting something other than “302 Found” it could be a Web Application Firewall triggering, or similar.
If you do need to share a screenshot be careful to make sure you aren’t sharing any cookie entries in the requests/response, as that could include the cookie that makes you WordPress admin (and you don’t want to share that to everyone).