IKA was one of a plethora of occupational insurance schemes in Greece. Each type of industry used to have its own scheme. IKA was the one for salaried workers so it was the biggest single one and it was the one that interfaced with NI/NHS in the UK to administer reciprocal benefits defined in EU Directives. If you were a salaried worker about 16 % of your gross salary was stopped at source and paid into IKA. The employer also had to contribute around 25 % of your salary. This is rather like NI in the UK although the percentages are bigger. Being insured with IKA provided you with healthcover and a pension on retirement like NI does in the UK.
Over time the government has gradually amalgamated most (all?) of the different schemes. Healthcare was the first to achieve "full" amalgamation and is now known as EOPYY. Slightly later pensions were also amalgamated and so the "full" insurance scheme is now known as EFKA. There may be a few schemes that are still not amalgamated, which is why I put "full" in quotation marks, but I am not certain. You will tend to see the terms IKA/EOPYY/EFKA bandied about indiscriminately.
Simply being a member of another country does not automatically grant you healthcare. For example if you are a UK citizen you only get healthcare in Greece if:
1 You are working in Greece and contributing to EFKA OR
2 You are a UK State Pensioner.
There are a few other odd, short term categories, like nannies, but the above are the main ones. If you are legally resident in Greece before the end of 2020, i.e. with a Residence Certificate, and, at some time in the future, you are entitled to a UK State Pension then you will become eligible for reciprocal healthcare through EOPYY when you claim your UK State Pension. Your wife will also become entitled as your dependent. You will need to ask the Overseas Healthcare Team for an S1 Form including your wife on it and you can use that to register at your nearest EOPYY office. The S1 Form means that EOPYY provide the healthcare but they recover the costs from the UK.
As Jeff says, since you will not be eligible for healthcover through an S1 form until November 2021, you will need some form of private healthcare cover for both yourself and your wife in order to get the Residence Certificate. You can get minimum cover that will satisfy the authorities relatively cheaply but it will only provide basic cover so it depends on your attitude to risk. Alternatively you can purchase fuller cover at a higher price. Stavros who posts on here can probably advise you, he is an insurance agent.
If you are getting an Irish passport through lineage and have never worked in Ireland I would be amazed if you can get an S1 form from the Irish government. I don't know how it works in Ireland but I would imagine they will only pay for your healthcover if you have contributed to some sort of national insurance scheme while working there.