I agree that it's really confusing. I actually never looked and saw the Assignment group field on the CIs. 

I've been looking around trying to see any other information about it, but most people seems as confused as you.

But I would reason like this:

Support Group: This is the group that supports the CI, which means that if there is e.g. an incident this is the group that should be put in Assignement group field on the incident.

Assignment Group: This is mainly just the group that the CI is assigned to. meaning that not all CIs is assigned to a specific user (Assigned to field), but has a group that it's assigned to. 

I would also say that in the majority of the cases, the group in these field would be the same thou and not sure if it's worth the effort to keep them up to date for this.

The above is still just my personal thoughts and ideas.

//Göran