When placing a Note with a Text Frame surrounding the text used for the note various options are available to define the look and sizing of the text and frame. This is all included in the Dimension Styles settings for Notes. One option that is not available is the capability to fill in that Text Frame with a background colour. These are a couple of options to be able to place these types of Notes.
Place a note with the Text Frame using one of the various bounding boxes. In this case I have selected a capsule.
Use the Create Region tool with Flood Fill, selecting a Fill Color and using an Associative Region.
This will place the background color within the frame so it will move with note.
Define a new Text Style that has the Background option selected along with a Fill Color and the Background Border offsets and symbology selected.
When placing the Note this time do not select a Text Frame as this is being defined within the Text Style.
The main limitation on this method would be with the selection of different styles of Text Frame like Capsules, Hexagons etc.
There will also be issues when using multiple lines of text or text nodes as the boundary and fill will only work on the separate lines.
Instead of defining the border within the Text Style you would have this defined in the Dimension Style. Note that you would have the Background Border color the same as the Fill Color in the Text Style.
This works a little better than within the Text Style with multiple lines of text but the background color still only covers the separate lines.