I gamble via my laptop rather than mobile.
Does those push notifications mean that Google has linked somehow my laptop and Gmail account?
What is the best way to protect myself from such "guardianship" (if any)?
Are using an android device or iPhone? If you are using an iPhone, I'm not sure these notifications would be coming to your phone because if there is something iPhone take seriously, it's users data and personalization of information, the only thing I think you will see is email messages. However, if you are using android device, then there is high change Google has personalized your information and they are giving you notifications based on what you have done before.
To avoid all these uncertain risk, don't use your important emails to registered into casino, used a good email but it shouldn't be the one that contain sensitive information on your phone, there is high possibility of casinos sending you marketing emails which you may not even want to see in your inbox and you can't be frown at them because you probably accepted their terms and conditions to use your email for marketing during the registration.