I tried to de-activate the Ajax Search Pro and then tried the log-in pop-up to log-in, but it's still not working – which means that the issue is very likely not related to this plugin.
What I can see on the console is a 401 error whenever an ajax request is sent to the WordPress admin-ajax.php handler, when the login button is clicked: https://i.imgur.com/IzzDnaj.png
This could indicate an invalid .htaccess rule that blocks the file, or a .htpassword protection on that folder, maybe an permission issue, or some way of protection over on that file. Perhaps this may help you to get it resolved.
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