'Tribal' isn't a very technical term and rarely means much. Many of the political entities of early medieval Wales originated as pre-Roman 'tribes', but this doesn't really tell us much about their political system. The use of the term 'feudalism' can be controversial, since some historians apply it far more specifically than others. It often isn't applied to early medieval societies because the warrior class was often directly maintained by rulers, rather than holding land in exchange for service. Early Welsh rulers had 'commanipulares', military companions who ate at their table, more like household knights than landed knights. We get this impression from the writings of Gildas and the poem Y Gododdin, which actually originates in the Welsh part of modern Scotland rather than Wales-proper, though it was preserved and revised in Gwynedd. This personal military retinue became known as a teulu.

There's relatively little evidence of Roman methods of government surviving into the early Middle Ages. Inscriptions show that Latin was used as a written language, but the Laws of Hywel Dda appear to have much less Roman influence than most European law-codes. The Welsh continued to afford some sort of significance to the Roman towns (see Gildas and the Historia Brittonum), but archaeology has demonstrated that the towns became massively depopulated at the end of the Roman period. Government in early medieval Wales was monarchical. It appears that they drew on Biblical examples of kingship (see David Petts, Christianity in Roman Britain), though it has also been suggested that there may also have been a folk memory of pre-Roman kingship. Wherever it came from, dynastic kingship was well-established by the sixth century.

The primary role of the king was that of a war-leader. Taliesin's poems for Urien of Rheged (a Welsh ruler thought to be from what became Cumbria) are a good source for this. They depict a ruler often at war with Anglo-Saxon kingdoms, who built up cattle-herds through raiding. Cattle was one of the main ways of holding wealth during this period. Poetic sources and stone inscriptions suggest that rulers presented themselves as defenders of their people, and viewed wars between Britons and Anglo-Saxons as ethnic conflicts, rather than very similar warrior-elites pillaging each other's lands (Welsh kings also fought each other). Indeed, Gildas's mention of a promise made by a king upon his accession suggests that there was some small element of 'government by consent', possibly originating with how fifth-century rulers were chosen by their soldiers. However, Gildas also indicates that rulers could get away with breaking such promises, so we can hardly call this 'constitutional monarchy'.