The first panel contains a picture of Jean-Luc Picard holding up a frame containing a number of medals. It has the text "These are your Starfleet merit badges, are they not, Mr. Data?"
The second panel has a picture of Data, with the text "Yes, Captain."
The third is a close image of the frame containing the medals. The medals take a variety of shapes, but the bottom-left medal resembles the Starfleet insignia. The panel has the text "And this shiny combadge in the lower left?"
The fourth panel is another image of Data, with the text "Communications merit badge, sir."
I think the joke is that the props department needed a collection of medals and one of them is just a combadge painted a different color. Maybe they ran out of ideas, maybe it was easy, maybe they assumed nobody would notice. Star Trek needed a lot of props that looked alien or futuristic but sometimes it's just a hair dryer with a curvy pipe glued on the outside and spray painted black.
Working the space phone isn't as anxiety ridden because they have absolutely no social obligations like being polite, saying a greeting, or even saying goodbye before hanging up. It's like the wild west of communication. Picard has literally muted people and then talked shit while they presumably could still see everyone on their bridge not paying attention to them.