Yes, the field keys need to be the same and the field types need to be the same across sites.
The reason that I know this is that I’ve had this issue myself, not with using update field, but with getting field values, but they are related. I had a field group on sub sites I was trying to get data from to show on the main site that did not work. The field group did not exist on the main site. I added the group to the main site set and everything started to work.
