5 and a half year old question, but with a thousand views, so it's clear that people still need this. I made a library which shows notifications using built-in Linux/MacOS/Windows notification managers: OsNotifications. You can just install it via NuGet:

dotnet add package OsNotifications

Currently it only shows very basic notifications, but it works on the three OSes mentioned before, which is apparently something no-one has done before (at least I couldn't find anything and this question hasn't been answered in more than 5 years). I will continue developing it according to the people's needs, so if anyone sees this and is missing a specific feature, just create an issue on GitHub.

This doesn't address the tray icon thing, but I hope this is better than nothing.