That is a very valid need. I have it myself. At the moment, based on observation and testing, it seems that multiple field groups can be assigned to blocks, but all the fields from all the field groups will appear in the same place, ie either in the content or the sidebar.
Perhaps the best way to assign the same field group to many blocks is by cloning the field group into the block’s fields. My need was to separate fields of content with fields that work like attributes to the content and have the first ones inside the content of the block editor and the latter ones in the sidebar.
