So, to begin with, the idea you are referring to in anthropology of "Band, Tribe, Chiefdom, State" is a model taken from Elman Service (who himself was working in the shadow of late 19th century Anthropologists with similar typologies), but this model has been largely abandoned by modern anthropology because it doesn't really suit a lot of examples. This is in part where some of the confusion comes from. Rarely does a society exactly fit one of these four "ideal types" of society, and so it can be extremely difficult to accurately classify societies in the past. For reference, here is a summary of Service's original typology.

I'm not well acquainted with Irish kings (hopefully someone else can address that point directly), but this is perhaps part of the issue, that these societies fit the model of states in some ways but also fit our model of chiefdoms in other ways. This doesn't mean this model is entirely useless. It does help us think about the tendancies of societies to organize themselves in particular ways, but we really shouldn't think of it as an ironclad rule that a society must fit into one of these boxes. Rather, it might actually be quite interesting to look at the instances where a society bucks the trend and doesn't fit neatly into one of those boxes.

As an example, Chaco Canyon in northern New Mexico is a series of archaeological sites (and a regional phenomena) that has elements from Service's typology of both a tribe, a chiefdom, and a state. On the one hand, it feels like a state or a very complex chiefdom. For instance, the massive structures ("Great Houses") built throughout the canyon and replicated in a huge area across the Four Corners. That speaks to something else that intuitively feels like a state or a chiefdom, the geographic spread of Chaco "culture" (in yellow on this map). Now, this doesn't mean that huge area was under direct political control of Chaco Canyon itself, but there is a significant degree to which the model established by Chaco Canyon was either mimicked or intentionally exported to areas outside the direct control of the Canyon itself, which is itself a kind of influence we might expect from complex societies (chiefdoms and states) even if it isn't direct political control.

Chaco Canyon is also home to some of the greatest diversity of objects in the entire prehistoric Southwest. We are talking about macaws and chocolate from Mesoamerica, potentially black drink from the Eastern Woodlands, and huge timbers imported from between 50 to more than 100 miles away. Alongside the obvious control of labor evidenced by the huge Great Houses, the "rulers" of Chaco Canyon clearly exerted significant social power.

However, this is where the issue lies. The evidence for social hierarchy and ranked lineages (two key elements of Service's definition of states and chiefdoms) is very rare at Chaco Canyon. At Pueblo Bonito - the largest of the Great Houses - there is some slight evidence of two different lineage groups, and of social hierarchy, but the evidence is somewhat ambiguous and we don't really see any of that evidence anywhere else in the Chacoan regional system.

Instead, we do see evidence for "ascribed achieved status", a key component of Service's typology. "Ascribed Achieved status" being that rulership is defined based on personal traits, rather than belonging to particular lineages or social institutions. We see this in modern Pueblo societies to a degree, where individual religious knowledge is one of the key factors dictating your place in society, rather than lineage. Your lineage may impact your opportunities to acquire restricted religious knowledge, but ultimately your social position isn't directly based on something outside of yourself as an individual. This probably was true in Chaco Canyon to a large degree, as well, even if lineage probably played some function as well. That we even have to debate how much these factors play into leadership in Chaco Canyon is itself testament to how much these supposedly key factors of a chiefdom or state are not at all obviously at play in what otherwise looks a lot like a complex chiefdom.

All this is to say that Service's typology (or any it was derived from or that it itself derives from) is not perfect and we shouldn't expect every society to fit into that framework. There is considerably ambiguity in how to fit certain societies into that framework, like Chaco Canyon with elements of tribes, chiefdoms, and states all at once.

There may be other factors at play as well. For instance, there is the consideration of how people described themselves. Again, without knowing anything about Irish kingdoms I can't say if this is true, but we might find a discrepancy between the actual political organization of a society and how the leadership of that society referred to itself. Leaders calling themselves "Kings" of what we would typically describe as a chiefdom.

On that note, you might be interested in this previous thread discussing "petty kingdoms" of Ireland.

You may also be interested in our Monday Methods threads, one on how to define a tribe and another on how to define Empire.

Edit: Mixed up "achieved" and "ascribed" status.