If you receive error messages from Google when trying to open the Google Webmasters Tools/Search Console integration, ensure that the account used to login to Google is the same account associated with the website property in Google Webmasters Tools/Search Console and that the domain specified is exactly the same one registered in Google Webmasters Tools/Search Console. Most importantly, the format of the domain must match the verification method, URL prefix or Domain DNS, that has been used to verify the property.
1) URL prefix method: in the case that the property has been verified using the 'URL prefix' method, the domain must match exactly including protocol and prefix.
So if you have registered the domain https://www.mydomain.com and you are requesting for example https://mydomain.com Google won't allow the access. The same happens if you have registered the domain https://www.mydomain.com and your are requesting for example http://www.mydomain.com.
In these cases you will receive the error message: '(403) User does not have sufficient permission for the site 'http://www.sitexyz.com/'
2) Domain DNS method: in the case that the property has been verified using the 'Domain DNS' method, the domain must be specified in the format: sc-domain:mydomain.com
Notice that in this case the prefix 'sc-domain:' is required before the domain part. Do not specify the protocol and the third-level part of the domain, indeed with this method your entire domain has been verified by Google Webmasters Tools/Search Console.
As visible below, it's possible to enter a specific domain in the component setting with the format for the URL prefix method:
or with the format for the Domain DNS method:
When you add a new property in the Google Search console, it's possible to choose the preferred verification method between 'URL prefix' and 'Domain DNS'. Based on the verification method that has been adopted, the domain must be entered in the correct format as explained above, otherwise the Search Console API won't work and you will continue to see the above error about missing permissions to access the requested site.