Ok...I solved my problem with the help of this Link: https://community.esri.com/t5/arcgis-pro-sdk-questions/daml-id-for-catalog-folder-context-menu/td-p/790840
This is the daml-text I used:
<updateModule refID="esri_geodatabase_module">
<menus>
<updateMenu refID="egdbItemFeatureClassTypeMenu">
<insertButton refID="ContextmenueChangeArcInfo_Menu1_Items_Button1" insert="after" placeWith="esri_projectItemPropertiesButton" />
</updateMenu>
</menus>
</updateModule>
Note that "ContextmenueChangeArcInfo_Menu1_Items_Button1" is the ID of a valid "ArcGIS Pro menu" which I added before.
Note also that this works only for "enterprise geodatabase feature classes and utility networks"
The best source of the "raw"-Ids can be found in the arc-gis-pro-git-lab here: https://github.com/esri/arcgis-pro-sdk/blob/master/DAML/ADGeoDatabase.daml.xml
After clicking the context menue my handler looks like this:
internal class Menu1_button1 : Button
{
protected override void OnClick()
{
var info = FrameworkApplication.ContextMenuDataContext as GDBItem;
MessageBox.Show("You chose the feature class: " + info.Name, FrameworkApplication.ActiveTool.Caption);
}
}