Is this the most piecemeal motorway in the UK? It's barely a mile long, but took about eight years to build in three different sections. Ouch.
Northern Ireland had grand motorway plans in the 1960s, with eight single-digit motorways radiating from Belfast throwing off spurs to various provincial towns. Aside from the M1 and M2, none of this happened. The M3 was initially to be the Belfast to Bangor motorway, but as this never got built, the new motorway section across the Lagan Bridge got the number instead.
The M3 is fairly obviously just the last mile of the M2, built in the 1990s when a new crossing over the Lagan was required. It is a continuation of the mainline of the M2 motorway and A2 dual carriageway, which means the three now form a route into the city from the north, looping back out to the north-east on the A2. Why exactly one small part of this route must be called M3 is beyond logical reason. Perhaps the route out to Bangor would have begun by crossing the Lagan here, but if you're only going to build a mile of it, messing about with a number seems picky.
Forgetting its bizarre numbering, the appearance of any new motorway in NI is a cause for celebration, and besides, nobody there actually pays any attention to its number anyway.