The Controls nuget package needs the tools\VisualStudioToolsManifest.xml file with the following contents:
<FileList>
<File Reference="Microsoft.Toolkit.Uwp.UI.Controls.dll">
<ToolboxItems VSCategory="UWP Community Toolkit" BlendCategory="UWP Community Toolkit">
<Item Type="Microsoft.Toolkit.Uwp.UI.Controls.AdaptiveGridView" />
<Item Type="Microsoft.Toolkit.Uwp.UI.Controls.BladeItem" />
<Item Type="Microsoft.Toolkit.Uwp.UI.Controls.BladeView" />
<Item Type="Microsoft.Toolkit.Uwp.UI.Controls.Carousel" />
<Item Type="Microsoft.Toolkit.Uwp.UI.Controls.CarouselPanel" />
<Item Type="Microsoft.Toolkit.Uwp.UI.Controls.DropShadowPanel" />
<Item Type="Microsoft.Toolkit.Uwp.UI.Controls.Expander" />
<Item Type="Microsoft.Toolkit.Uwp.UI.Controls.GridSplitter" />
<Item Type="Microsoft.Toolkit.Uwp.UI.Controls.HamburgerMenu" />
<Item Type="Microsoft.Toolkit.Uwp.UI.Controls.HeaderedTextBlock" />
<Item Type="Microsoft.Toolkit.Uwp.UI.Controls.ImageEx" />
<Item Type="Microsoft.Toolkit.Uwp.UI.Controls.ImageExBase" />
<Item Type="Microsoft.Toolkit.Uwp.UI.Controls.MarkdownTextBlock" />
<Item Type="Microsoft.Toolkit.Uwp.UI.Controls.MasterDetailsView" />
<Item Type="Microsoft.Toolkit.Uwp.UI.Controls.Menu" />
<Item Type="Microsoft.Toolkit.Uwp.UI.Controls.MenuItem" />
<Item Type="Microsoft.Toolkit.Uwp.UI.Controls.OrbitView" />
<Item Type="Microsoft.Toolkit.Uwp.UI.Controls.OrbitViewItem" />
<Item Type="Microsoft.Toolkit.Uwp.UI.Controls.OrbitViewPanel" />
<Item Type="Microsoft.Toolkit.Uwp.UI.Controls.PullToRefreshListView" />
<Item Type="Microsoft.Toolkit.Uwp.UI.Controls.RadialGauge" />
<Item Type="Microsoft.Toolkit.Uwp.UI.Controls.RadialProgressBar" />
<Item Type="Microsoft.Toolkit.Uwp.UI.Controls.RangeSelector" />
<Item Type="Microsoft.Toolkit.Uwp.UI.Controls.RotatorTile" />
<Item Type="Microsoft.Toolkit.Uwp.UI.Controls.RoundImageEx" />
<Item Type="Microsoft.Toolkit.Uwp.UI.Controls.ScrollHeader" />
<Item Type="Microsoft.Toolkit.Uwp.UI.Controls.SlidableListItem" />
<Item Type="Microsoft.Toolkit.Uwp.UI.Controls.SurfaceDialTextboxHelper" />
<Item Type="Microsoft.Toolkit.Uwp.UI.Controls.TextToolbar" />
<Item Type="Microsoft.Toolkit.Uwp.UI.Controls.TileControl" />
<Item Type="Microsoft.Toolkit.Uwp.UI.Controls.WrapPanel" />
</ToolboxItems>
</File>
</FileList>
(Note: We should probably exclude the deprecated controls #1465)

issue closed because PR is merged.
Will it work for non-UWP projects?
All controls are built for UWP, they wouldn't work in other type of projects.
I mean will this approach work for other types of project? For example, if I create an package with ASP.NET controls inside.
Ah, good question. I believe it would but don't know for sure. @dotMorten might know
It works for WPF, so it might work for ASP.NET. @diverdan92 might be able to confirm, (or you could just try it out :)
No, it will only work for XAML controls shipped in NuGet packages (UWP + WPF).
...this is where you add "...but we're working on it" :-)
Most helpful comment
...this is where you add "...but we're working on it" :-)