"Is Éireannach mé" is a copular sentence, so it's technically "I'm an Irishman" (or Irishwoman) - Éireannach is a noun in this case, whereas "Irish" is an adjective in "I'm Irish", and Éireannach is also the form of the adjective - "tá fuil Éireannach aige" - "he has Irish blood/descent"
English uses the same adjective to refer to languages and to the people and products of a country. Irish uses two different words to refer to the language of a country and to the people and products of a country- Gearmánach is used to describe things from Germany, Gearmáinis is used to describe things about the language. So you would use a leabhar Gearmáinise to learn German, but you might drive a carr Gearmánach. You might use a foclóir Fraincise, but watch a scannán Francach.
In the same way, Éireannach is "Irish" in the sense of from or about Ireland, whereas Gaeilge refers to the Irish language.
Someone who holds an Irish passport can probably make a reasonable claim to the term Éireannach. I'm not sure that every passport holder could reasonably be called a Gael, though.
But as with any discussion about "who is Irish", the answer is often in the eye of the beholder.
Gael would be ethnicity, Éireannach would be nationality. Someone from the west of Scotland might be a Gael without being Éireannach. A Pole who has settled in Ireland and gained citizenship would be Éireannach, without being a Gael.
In general we would always use Éireannach. There are rarely non-sinister reasons for explicitly pointing out ethnicity, when you are the majority ethnic group. In other words it can come across as "I'm real Irish, you are not", which would be a horrible message. You might see it with an Irish and a Scot bonding over shared heritage in an inclusive way.
Éireannach would be the more appropriate, inclusive and far more common usage. "Is Gael mé" sounds a little like someone from England introducing themselves as "Anglo-Saxon". It isn't technically incorrect, but it's a little weird.