Do Private Windows make me anonymous to my service provider?

Whoever connects you to the Internet (your ISP) can see all of your network activity even while in a Private Window. If you are on mobile, this is often your mobile carrier. If you're connected to a WiFi network, this is the owner of that network and if you're using a VPN, then it's whoever runs that VPN.

Your ISP can see which sites you visit as you visit them. If those sites use HTTPS, they won't be able to do much more than make an educated guess about what you do on those sites. If a site only uses HTTP then your ISP can see everything you do: your search terms, which pages you read, and which links you follow.

If an employer manages your device, they might also keep track of what you do with it. Using Private Windows probably won't stop them from knowing which sites you've visited. Someone else with access to your device could also have installed software which monitors your activity, and Private Windows can't protect you from this either.