I connect here: \IPS\Theme\class_core_front_global

Hook:

CSS Selector: #elSignIn_submit

Code:

{{$formcaptcha     = new \IPS\Helpers\Form\Captcha;}}
{$formcaptcha|raw}

If I enable the hook on keycaptcha, then on the Sing UP form - keycaptcha stops displaying. What could be the problem?

The error is indicating the URL to your website is incorrect in your KeyCAPTCHA control panel. Have you logged in there to verify it is set correctly?