[See Estimating the proportion of Puritan genes in America's white population for links to census data.]
"A Survey of Irish Surnames 1992-97" (pdf) lists the following as the 10 most common surnames in Ireland in the 1990s:
1. Murphy 2. (O)Kelly 3. Walsh(e) 4. (O)Connor 5. (O)Sullivan 6. (O)Byrne 7. (O)Brien 8. Ryan 9. Smith/Smyth 10. (O)Neill
We'll exclude Smith/Smyth for obvious reasons. The remaining 9 most common names, all of Gaelic origin, cover 7.85% of the 1990s Irish population. (With the 1890 data, the number would be 7.67%; but that's leaving out some of the variants included in the 1990s survey.) Northern Ireland's inclusion in the survey might end up inflating our surname-based Irish Catholic population estimates by something like 10%, but I'm not worried about this level of error right now.
The number of US whites bearing one of the nine most common Irish surnames in 2000, from Census data: 1188571
The extrapolated equivalent total number of Irish individuals among the US white population in 2000: 15141032
Which comes out to 7.78% of the ancestry of the US non-Hispanic white population in 2000.
15 million (or maybe 13.5 million) descendants is certainly a more plausible biological outcome of 4.5 million Irish immigrants than the "40 million Irish Americans" we see from census self-identifications.
But it appears there's considerably less disconnect between levels of Irish ancestry and Irish self-identification in Massachusetts (vs. the US as a whole).
In the 1940 Census (the 2000 Census surname data is not available broken down by state), 87028 Massachusetts whites had one of the nine most common Irish names. Based on that, we can estimate the number of Irish in MA was 1108637 -- or 25.9% of the total 1940 MA white population of 4280019.
The 2005-2009 American Community Survey 5-Year Estimates put the Irish proportion of the Massachusetts population, based on self-identification, at 23.7% (vs. 11.9% for English). Or, considering only the non-Hispanic white population, something like 29% identify as Irish.
This better agreement likely reflects relatively lower levels of intermarriage in MA, as might be expected from the state's greater Irish concentration.