I Believe this is not just yoit standard double Irish, there was an extra loophole that Eire closed in 2014 where the second company could be based in no country at all, not just a fiscal haven
Unless you're speaking Irish, please don't refer to the country as 'Eire' - it's akin to referring to Germany as 'Deutschland' in English. It's either 'Ireland' (the country's name in English) or 'Republic of Ireland' (its official description) if you're more comfortable with that.
But, but... Isn't the modern norm to use "native" names wherever possible? Calcutta/Kolkhata, Peking/Beijing and so on? As a token of respect towards actual indigenous cultures.
No. This predates that by decades. Also, our constitution clearly states the specific name of the country in English and Irish. This might seem like a small thing, but it' actually a very important issue to Irish people: https://en.wikipedia.org/wiki/Names_of_the_Irish_state
It's intended as quite the opposite of a token of respect: the purpose of referring to Ireland as 'Eire' (omitting the accent on the initial 'é' too) is to delegitimise the use of the term 'Ireland' to refer to the Irish state.
If actual Irish people wanted the state to be referred to as 'Éire' in English, it'd be one thing, but we don't. In English, it's name is 'Ireland', and in Irish, it's name is 'Éire'. There's also a diplomatic fudge in that there's an 'official description of the state', which is 'Republic of Ireland', which isn't its official name, but which is an acceptable substitute. And yet, incorrect terminology is still used to refer to the Irish state, such as 'Éire' (being used in English) or 'Irish Republic', and sometimes you find particularly ignorant types refer to it as 'Southern Ireland' (which was the name of a failed attempt at a counterpart to Northern Ireland within the UK, but which never gained any legitimacy) and 'The Freestate' (which carries the implication of British dominion over Ireland).
The norm is to use 'native' names where those people want those names to be used. Both Ireland and Éire are native names of Ireland, just in two different languages.
And that has long stuck in the craw of Irish people. It's seen as a way to subvert the legitimacy of the Irish state here in Ireland when the country is referred to as 'Eire'in English.