According to official stats, 16% of Irish residents are citizens of other countries. Keep in mind that this number will exclude foreign nationals that got Irish citizenship through naturalization (and therefore became Irish citizens).
https://en.wikipedia.org/wiki/Foreign-born_population_of_the...
Most recent numbers from the UK list 16% of the population being "foreign-born". While this number may be similar to Ireland, it still counts someone as foreign born even if they became UK citizens by naturalization.
Also, consider that one of the most prominent migration sources for the UK is of Irish nationals (that can live and work in the UK even after brexit). Irish culture is not too dissimilar to UK culture (especially considering that Northern Ireland is currently part of the UK).
If anything, Ireland experienced more foreign culture immigration than the UK, not less.
Your point is invalid.