Can I snooze notifications from the browser on High Sierra?
Safari doesn't give you the option to exactly "snooze" website notifications (because the snooze feature is Siri suggested with that notification), but you can manage them:


Stop receiving notifications from a website: Choose Safari > Preferences, click Websites, then click Notifications. Find the website in the list, then select Deny.
Change how website notifications appear: Choose Apple menu > System Preferences, click Notifications, select the website on the left, then change the notification alert style and other settings on the right.
Stop seeing notification permission requests in Safari: Choose Safari > Preferences, click Websites, then click Notifications. Deselect “Allow websites to ask for permission to send push notifications.” From now on, when you visit a website that can send you notifications, you aren’t asked.


Perhaps when MacOS Mojave goes public we will see that as an option. If not, feel free to join the beta/development team and petition your request for that feature. 

In the mean time these options work best.