You are over thinking this.
If you dont feel Irish, and you are not aware* of your genetics, and you live here in the US - how COULD you be Irish?
*unawareness is key here
Fat, black, Chinese(arguably), female, male, short, fat are objectively discerned
Irish ethnicity is not. I have no idea what an Irish-American person looks like that is different from, let say, a Fin or a Pole.
I do. But then again I am one.
Look at the Kennedys and their big heads - Irish! Plus Ted was a drunk in his younger days.
There are markers, though as I said an inexact science.
The DNA test is not exact either - more so for for African, Indian and Jewish ancestry.
Further information: Genetic history of Europe
For people with European maternal ancestry, mtDNA tests are offered to determine which of eight European maternal "clans" the direct-line maternal ancestor belonged to. This mtDNA haplotype test was popularized in the book The Seven Daughters of Eve.
SNP testing may enable mostly European individuals to determine to which Sub-European population they belong:
Northern European subgroup (NOR)- mostly Northern and Southwestern European
Southeastern European (Mediterranean) subgroup (MED)- mostly Southeastern Europeans (Greeks, Albanians or Turks)
Middle Eastern subgroup (MIDEAS)- mostly Middle Eastern
South Asian subgroup (SA)- mostly South Asian from the Indian sub-continent (i.e. Indian)