If the text formatting is coming from the underlying paragraph style then you won't get it but local character formatting will be preserved by this modification.
Code:
Sub Macro1()
Dim oRng As Range, oFN As Footnote, rngInner As Range
Set oRng = ActiveDocument.Range
With oRng.Find
Do While .Execute(FindText:="\[\[(*)\]\]", MatchWildcards:=True)
Set oFN = ActiveDocument.Footnotes.Add(oRng, , oRng.Text)
Set rngInner = oRng.Duplicate
rngInner.MoveStart Unit:=wdCharacter, Count:=2
rngInner.MoveEnd Unit:=wdCharacter, Count:=-2
oFN.Range.FormattedText = rngInner.FormattedText
oRng.Text = ""
Loop
End With
lbl_Exit:
Set oRng = Nothing
Set rngInner = Nothing
Exit Sub
End Sub