It's certainly got a lot in common but I think Overground generally acts a distributor across more suburban, less densely populated areas. The service, again generally, being less frequent, slower, and yes running overground, reflects this.
The Elizabeth line on the other hand is a full scale, long distance, commuter railway. It also is underground within zone 2. And yes the closest to this is Thameslink.
Personally I'd love to see Thameslink come under the TfL umbrella. The Crossrail brand could then be revived for the mode, with the Elizabeth line and Thames line being it's services.