In the Builder, it is impossible. By coding, you can easily generate the code of the Field Group and copy the code which created the field and paste to other places as you want. I also note this case and discuss with the developer team to support move/copy a field to other Field Groups.
I agree this would be really useful.
In the meantime, to move fields that already have data into new groups, I'd just like to check.
Will this process ensure no data is lost?
create new custom field with same field type, settings and id inside the group
The data will be lost because the field ID is store in the database as a meta key. When you create a new field in a group field, the meta key is group ID so you have to add the field value again.
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