I'm not sure which marketing department you mean? MonoGame doesn't have one
MonoGame is an open source port of XNA which allowed you to write games for Xbox 360, PC and WIndows Phone 7/8 - in addition Microsoft provided Xbox Live Indie games as a store front for XNA games on the 360. On PC you had to make your own arrangements. Xna is a technology on life support at this point... the SDKs are 5+ years old, have trouble installing on windows 10 and the XBLIG program is due to shut down for good in November.
MonoGame supports PC though Win32 (traditional windows app) and UWP (windows store apps). It does not support XBox 360.
Recently Microsoft has announced the ability to run UWP apps on Xbox One https://docs.microsoft.com/en-us/windows/uwp/xbox-apps/ though there's been no specific mention of UWP on XBox One X Microsoft has said they are all compatible so there's no reason to believe that a UWP app will not run on the original, the S and the X variants.
You may publish your UWP app in the windows store, you can apply to the ID@Xbox program for an official Indie release or you can target the very new creators program which is in preview now https://developer.microsoft.com/en-us/games/xbox/xboxlive/creator - you will note a mention of MonoGame on that page...
Will we be able to tap any special Xbox One X resources? Right now anyone who knows and has a dev kit would be under way but as a beginner thats likely not going to be an issue for a while.
I recommend you make UWP apps for windows, get them published in the WIndows store and test them very well for controller support - you will likely find they port over to the Xbox almost trivially?