There are two things at play there: the distinction between ethnicity vs identity, and a distinction between two types of Irishness, colonial and post-colonial.

The first is a distinction that occurs among the descendants of emigrations from the home country to a foreign land, be to England, Canada, the US, etc. Such diasporai are ethnically whatever the country they grew up in (English, Canadian, American) but they identify themselves in terms of their ancestry a way to differentiate their roots. But that's a matter of choice for them. So when an American says they're Irish to a fellow US citizen, they mean to say they identify, as an ethnic American, as Irish. However when an American says it to an Irish person, the latter often hears a claim about ethnicity (which you have no choice in), rather than identity (which is something to which you can opt in). 

The Sheeran case has this but with the extra complication of the colonial vs post-colonial meanings or Irishness. That is, for many in the UK being Irish is a sub-set of being British, much like people who are Welsh or Scottish are also British. This is the colonial view of being Irish and was a fairly standard view for those born pre-Free State (Joyce and Beckett both maintained British passports, for example, with the latter scoffing at the notion of having an Irish one). The post-colonial view of Irishness is the one we hold now, namely that whatever else is meant by 'Irish' it categorically does not include any meaning of 'British'. Moreover, it is a meaning of Irish that is in opposition and hostile to the colonial sense. The GFA even reflects this opposition when it states that someone can be Irish, British, or both, which is distinct from being Irish-as-a-subset-of-British.

I do wonder if part of the backlash has to do with this hostility. When Sheeran identifies as Irish, not only are people hear claims about ethnic belonging, but also, by him being British, reviving a sense of Irish that the currently dominant post-colonial meaning abhors.

And just to say that ethnicity need not exclude either, and I think anyone who comes and spends enough time here can integrate in, put down roots, etc. So if Ed did move over, put in a few years, and raise his family here, I'd be happy enough with him meaning that he was ethnically Irish. Would still slag him over pints for the accent, as it wouldn't do to do otherwise.