The best example I heard from a friend who was a recent immigrant from India.

He had quite a few friends who were the children of immigrants who arrived from India a generation earlier.

When my friend’s parents visited (from India where they still lived) they were shocked when these 2nd generation American-India kids greeted them in a super formal way that had disappeared in India decades ago.

Basically the immigrants to the US held onto old customs more strongly than the ones who never left India. And they passed that onto their kids. So they were “more Indian than people from India”.

The same is true for German immigrants in South America.

I suppose, in a way, similar for the Amish in the US.

Speaking the old language and maintaining a 1700s lifestyle, for the most part.

The "more Irish than the Irish themselves" phrase originated as a description of mediaeval Irish history. Norman-ruled England conquered Ireland, and introduced Anglo-Norman settlers – Irish society became divided into two social castes, an English/French-speaking ruling caste and a subordinated caste of indigenous Irish-speakers. (This was prior to the Protestant Reformation, so everyone involved was Catholic – indeed, the English monarchy's conquest of Ireland was approved by the Pope.) The English became concerned that, after a few generations, the Anglo-Norman upper caste began to speak Irish and intermarry with the native Irish-speakers – they saw (quite accurately) that this would lead to weakening of English rule and eventual demand for independence. "more Irish than the Irish themselves" was really meant as a somewhat hyperbolic/ironic reference to this process of cultural assimilation. The English responded with anti-Irish legislation, prohibiting the speaking of Irish, intermarriage between English-speakers and Irish-speakers, and formally subordinating the theoretically independent Irish Parliament to that of England – however, the legislation largely failed to be enforced, the blurring of the boundaries between the native Irish and the Anglo-Norman newcomers continued, and English rule became (in much of the country, especially the parts furthest from Dublin) more theory than fact, as local lords found they could basically ignore the edicts of the English administration in Dublin and do whatever they liked.

This is arguably the earliest historical roots of the Northern Ireland conflict, although with various other layers added on top – the introduction of a religious dimension to what was originally a purely cultural/ethnic/linguistic/political conflict due to the Protestant Reformation; further waves of settlement from Britain; the Irish theatre of the English Civil War and the later Jacobite-Williamite War in which many Irish (especially, but not exclusively, Catholics and Irish-speakers) supported the deposed King James II against the regime of William of Orange.

I don't know how much sense the phrase has in the context of immigrants to contemporary Ireland. I suppose some users of it must have found some way to connect it to contemporary affairs–probably there is some immigrant somewhere who is obsessed with teaching their children to speak Irish when the majority of Irish people don't make much of an effort to do so themselves–but I think some use of it may also be motivated by its long history and memorable phrasing rather than genuine contemporary applicability.