Not necessarily. In the case of link roads a route number can be shown without brackets - see the "Proximity Rule" in Chapter 2 of the Traffic Signs Manual. While it refers to local roads specifically, I can see it being construed for use here (which is why heading towards the M50 signs indicate so, and vice versa for the R139, though I'm led to believe the blue route confirmatory signs and the sign for the Clonshaugh Rd roundabout are a mistake).

As for the motorway link, is there somewhere where it is specifically stated that they indicate motorway restrictions in place? My understanding of the tag from the OSM Wiki is that it isn't necessary the case.

Then I am of the opinion that the route on both sides should be classified as highway=motorway link. That is the purpose of the road. It is done in this way for the M3 link roads at Navan and Kells, which are also legally classified in the same manner as this stretch of road.

What do you mean by "old paperwork"? If by "ground truth" you mean the N1 signage, by this logic the whole R132 from Shantalla Bridge to the city centre should also be marked N1 as there still remains plenty of outdated signage indicating N1.

The Classification of Regional Roads clearly states:

R139

"Clonshaugh — Baldoyle, Dublin (Part of old National Route 32)

Between its junction with M50 link road at Clonshaugh Road in the county of Fingal [...]"

Clonshaugh Road is not at the Turnapin Interchange, but at the subsequent roundabout. The M50 link road therefore has no number.