Quite simply, we don't know. Scholars have long postulated that Constantine III would have taken the comitatenses (field army) of the comes brittaniarum (Esmonde-Cleary's The Ending of Roman Britain includes a widely-known example of this). The problem is, the Notitia Dignitatum doesn't give us any idea as to how large this force was. As for Britain's limitanei (less well-equipped garrison troops), current scholarship is moving towards the idea that most of them remained in Britain, having become embedded into local communities (see Rob Collins, *Hadrian's Wall and the End of Empire). How Roman or British Constantine III's army was in character is, again, unclear. Stuart Laycock has argued that the large number of British military belt buckles found at Richborough indicates a large number of native irregulars embarking for Gaul. Though we can't necessarily associate this with Constantine III.

Yes, they would have been wearing similar clothing and armour to Honorius's troops. There's nothing written about whether Constantine III used a distinct banner. However, this kind of homogeneity was fairly common in pre-modern warfare. Roman armies fought each other nearly as much as they fought barbarians. There's all sorts of speculation on how soldiers in such armies recognised their own side and the enemy, and it's not really within my remit here. Though I will mention the idea that people typically fought alongside people they knew, so they would have often been able to differentiate between sides simply based on facial recognition.