I observe that Rep. Ireland is a notable absentee from the group, despite being one of our 'Anglo Mates', and sharing the same views regarding Global security. EU membership aside, why is this?
Ireland has had a policy of neutrality as far as international relations are concerned since WWII. Although this has never been formally codified in their constitution, and indeed recently such an amendment was rejected, neutrality is an important part of Irish foreign policy which has also precluded the country from joining organisations such as NATO, where it would seem a clear candidate.
The population of Ireland has generally remained extremely supportive of the policy of neutrality, a 1996 poll showed 99% of respondents supported Ireland maintaining the policy. It seems likely that had a governing party sought to join the military alliance, this would have translated to a drastic fall in public support upon the decision being revealed.
Finally, the Five Eyes alliance can trace its origins back to the post-WWII Atlantic Charter which set out the goals of the Allied forces in the post-war era. As Ireland remained neutral during the war, they were not directly involved in these discussions.
and sharing the same views regarding Global security.
I think this is a fundamental flaw in your viewpoint. Ireland has demonstrated a very different viewpoint from the UK and US and their primary allies. This goes beyond any historic differences on Irish-centric security and derives primarily from Ireland very much seeing itself primarily as a European state.
Ireland actively participates on European security issues (constitution allowing, see CDJB's answer for more), whereas in no way could the US, Australian, New Zealand, Canada or even the UK (which has formally left the EU) been said to be focused on EU security matters. I think it's fair to say that the EU clashes with these countries as often as not on security issues. Ireland is a founding member of the Common Security and Defense Policy group, further demonstrating Ireland's commitment to the EU.
Ireland has a very strong connection to European politics and all other ties do seem to be weaker.
Irish neutrality in WWII was a different issue and I do not think, in the modern context, it relates to this issue. Ireland that time was a poor country coming out of a very difficult Civil War (1920's) and political views at that time, while not being pro-German, were also definitely not pro-UK. At that time Ireland's European ties were quite weak, so the primary focus was on avoiding further conflict where possible. After WWII Ireland's focus was on development and when (in the 1970s) Ireland was in a position to join the EEC, Ireland as a nation embraced this opportunity in a way that our UK neighbors never really did. Ireland adopted it's new European identity as it's primary one, whereas the members of Five Eyes do not. So while we share some common viewpoints with Five Eyes, we could not be said to be a "good fit" to it, any more than other EU members are.
FVEY developed from the second world war and the intelligence sharing between various allies during and after the war.
Ireland was not allied to the UK and the USA during the war (it remained technically neutral) Moreover Ireland was not part of the Commonwealth, and the "five eyes" club was an agreement to share intelligence between the US and the British, including aligned members of the Commonwealth.
The important thing to recognize is that while Britain may think of Ireland as a "mate", Ireland thinks of the British state as the former colonial power from which it won independence violently, which partitioned Ireland, and which ran a policy of state violence against Irish Catholics in Northern Ireland for decades until the peace treaty in 1998. (A plan was even drawn up for armed intervention by the Republic in 1970, although only for the purposes of recognizing that it was infeasible)
Ireland itself has very little in the way of state security apparatus, or even military, but it does have a small Department of Military Intelligence.