There is some confusion here between citizenship and passports. You can be an Irish citizen without a passport. I’m currently both a British and an Irish citizen despite not having a valid passport.
The first thing you need, if not born on the island of Ireland, or born to a parent born etc, is to be on the Foreign Births Register. I think you can only be on that if you have a parent who was a citizen when you were born. There’s a nominal cost to this. Once registered, any children you have post-registration can also be registered.
As mentioned above, the list of signatories is stricter than for U.K. And be aware that both the Foreign Births Register and passport office do call your signatories to check they’re real – happened when registering my son and his first passport; and an uncle had his passport application rejected because his dentist was on holiday when they called (a few weeks, I don’t know how often they try).
There is nothing in either UK or Irish law to prevent you having both citizenships and both passports.