No. The UK does not recognise the EU as a sovereign state but as a multinational organisation and has offered to treat it's delegates appropriately. As is standard under the law. Just the same as NATO etc are treated.

If the EU wishes to be treated differently to other multinational institutions then it can negotiate an exemption to be taken through Parliament. It hasn't done so yet. Obviously wasn't a priority for Barnier - he was aware of this at the time and didn't deal with it.

If he'd negotiated as part of the agreement that the UK would give the EU full diplomatic status and included that in page 1188 of the agreement then would anyone have noticed? Or cared? But he didn't.