I’ve added the proper keys on the backend but the reCaptcha part does not even show on the contact form
Does the form still let you submit, or is there an error?
If it still submits, the captcha is likely not being added at all. If there’s an error, then the captcha is probably getting hidden, possibly by a browser adblocker, or maybe custom styles.
OK, so the problem here is that you’re using a pretty old version of Omeka Classic, and Google just doesn’t run the version of reCaptcha that version of Omeka supports anymore.
Omeka 2.6, released in 2018, switched to the newer “v2” reCaptcha. To have it working you’d need to upgrade at least to that version (though I’d probably recommend using the latest instead).