Would this be because if it is played across the rows it would have the sharps and flats on the C# row and the 'white' notes on the C row? And therefore would be chromatic?
That's exactly right, Kiwi. This is one of several tunings in which the two rows are a semitone apart. Apparently the system was invented in the 1880s by an English concertina and melodeon maker, George Jones.
These days "half-step" system boxes are mainly played by Irish and Scottish musicians. C/C# has gone out of favour - the predominant system for the past 50-60 years has been B/C, with C#/D making a strong comeback more recently. In these systems, unlike the C/C# (and another older system, D/D#) , the inner row is the main one.
These boxes are chromatic in that you have every note in the chromatic scale. But playing in keys that are very remote from the key of the main row is difficult. Most players don't go beyond a fairly small range of keys - quite a few more than are available on a D/G or other "quint" box, though. And you're never stuck for accidentals. If you wanted to learn this system, esp. for Irish or Scottish music, you might want to think about tuning it down a semitone to B/C or up a semitone to C#/D.
Cheers - and welcome to the forum!
Steve