There was actually a very heated debate in the 1990's archaeological world about whether the word Celt/Celtic is appropriate to describe the Iron Age peoples of Ireland and Britain and whether there ever existed a group of people who ethnically identified themselves as "Celts" at all anywhere. It got rather nasty with some scholars on the celto-skeptic side being accused of the erasure of the culture of historically oppressed peoples (Irish and Scottish people).

The short answer to your question is no, the medieval Irish literature never identifies Irish people as Celts or Celtic.

The long answer is that the division of pre-Roman conquest western Europe into Celts and Germans in modern history and archaeology is due to the methods of archaeologists and classicists in the 19th and 20th centuries. People in the past were lumped into "Cultures" based on particular suites of artefacts (or sometimes one type of artefact) that were found in archaeological sites. For example, in the Bronze Age, the "Beaker people" spread to Britain and Ireland, identified by a particular ceramic Beaker. What was even better was when you had in the histories of the Classical world a mention of the peoples they came into contact with, telling you what they were called, where they lived and what they were like. For example, Julius Caesar says that the people who live in what is now modern France are called "Gauls" by the Romans but "In their own language they are called Celts". This was great for the 19th and 20th century archaeological enthusiast because they could look at what artefactual evidence there was from France in the immediate pre Roman conquest period and match that up with the people named by Caesar. Now whenever someone excavated these types of artefacts, whether in Ireland, Germany or France, they knew that the people who lived in that place were Celts! Simple!

Or not so simple.

The standard pre-1990's understanding of the Celts was that they emerged in Central Europe (South Germany, Austria, Switzerland, East France) and spread through migration to South Eastern Europe, down to the rest of France, into Britain and Ireland, and into Greece and Italy where they clashed with the Classical civilisations who gave us our modern idea of blonde spiky-haired warriors whose bravery was without question but lacked the tactical cohesion to overcome the regimented armies of the Greeks and Romans. But this is based on finds of archaeological artefacts that all belong to the same art style, the La Tene style. Finds of artefacts in this style were used in conjunction with the classical sources and linguistics (the Gaulish, Irish, and Welsh languages are all part of the same language family), to appoint the people of these places as "Celts" both past and present.

For Ireland in particular, because I'm an Irish archaeologist and know that stuff the best, there is zero evidence of a mass movement of people from the continent to Ireland in the Iron Age. La Tene art appears in Ireland in roughly 300BC but it is a local variation rather than the wholesale transportation of continental or British artefacts to the island, making it unlikely that there was a small invasion that displaced the local rulers but left the majority of the population as was (think the Norman invasion of Britain for this kind of Elite takeover). Ireland remains peculiar in a lot of ways compared to Britain and the continent throughout the Iron Age and scholarly consensus since roughly the 1990's has been against a movement of people (no matter what you decide to call them) to Ireland at this time.

As for why the idea of Ireland as Celtic is so enduring in public imagination, that is an entirely different question and answer.