Identity is the funniest thing. One of the funniest things about American identity is that the percentage of people claiming to be "Irish American" is far higher than the percentage that it demographically should be, while the percentage of people claiming to be "English American" is far far lower! I suppose this is because Ireland has more friendly and exotic connotations than England does; in any case, it shows that identity does not necessary reflect culture or blood or ethnicity, but instead a desire for exoticism or culture or nostalgia.
In Britain there is less of a tendency for people to identify with ethnicity/nationality of their forefathers if their forefathers are from overseas. Both my mother's parents for example are Irish and moved to Britain as adults, but it just would not occur to me to describe myself as Irish or from Ireland. The expression "Irish British" just does not exist. Irish friends of mine would think me ridiculous of I described myself as Irish given that I have only been there on holiday and culturally am British, but I might mention in passing that my grandparents are Irish.
My experience with Australians is that they all describe themselves as Australian and are quite reluctant to describe ethnicity in more detail than that. I have quite the opposite experience with Americans who are all keen to indicate some European heritage when they speak to me (but never British heritage!). I don't mind too much, but I do find it a bit peculiar and I think it is strange unless you have some real and powerful connection with a foreign (i.e. you spoke Polish at home or something rather than celebrated Saint Patrick's day).
Essentially I don't agree that geneaology equals identity.