I don't think pop singers can be described as Tenors in the context, even if they sing in the tenor range and have very good voices.

A classically trained singer or a singer influenced by the classical style is what is referred to when talking about Irish Tenors. They may sing a folk song with an Irish accent and lighten the tone a little but the style is still very unique.

Irish folk or trad singers never sing in the Irish Tenor style.

With all due respect to Billy Joel, I don't think he really knew what the term Irish Tenor meant.