Dear Mr Bowey,
Thank you for your question. I am afraid that you have not received quite the correct and full information.
The possibility to travel freely without visas and settle in the EU, for your wife, is limited to the following situation:
a) she has a residence card as a spouse of an EU citizen, AND NOT simply based on long-term residence;
b) she has a special visa (see below); or
c) she is exempted from visa based on her passport (which does not appear to be the case, given that she is a Chinese citizen), in which case, with your birth certificate, she can benefit from the right to settle.
To clarify this a little, she would have the right to travel freely if she would have the residence permit for family members of EU nationals described here: (https://www.gov.uk/apply-for-a-uk-residence-card/eligibility). I am quite certain that, at this point in time she cannot obtain such a residence card because in the eyes of both Irish and UK law, you are not an 'EU citizen' but a local citizen, for whom local rules apply. Thus, the obligation to go through the FLR Resident Permit.
She would be able to obtain such a Family Member Card from any other EU country than the UK or Ireland. An example, for France, is here: (https://www.service-public.fr/particuliers/vosdroits/F13512).
Regarding the special visa, which is what I recommend, I must inform you that as a married couple, she can apply for a visa for any Schengen country (let's say France), which is then valid for any other EU country, and this visa is free of charge and granted automatically. For these EU countries, you would be an 'EU citizen', and therefore you would then qualify to take her and reside or just travel if you wish. This is a special visa for non-EU family members of EU citizens.
When requesting the visa, you would have to present the marriage certificate. And, to answer your initial question, the marriage certificate does not have to go through any particular process to be recognized in the EU, because there is no such process. Each country has its own recognition procedure. For Chinese (PRC) documents, unless they are issued in HK or Macao, they need to be "Legalized". This means obtaining the following stamps, in the following order:
1. Stamp of the Chinese Ministry of Foreign Affairs;
2. Stamp of the Beijing Embassy of the country where you wish to use the certificate;
3. Stamp of the Ministry of Foreign Affairs of the country where you wish to use the certificate.
For documents issued in HK or Macao, a simpler procedure, called the Apostille, is available. This is a stamp applied by local authorities, namely for HK here (https://www.hcch.net/en/states/authorities/details3/?aid=633) and for Macao here (https://www.hcch.net/en/states/authorities/details3/?aid=634). With the Apostille, the certificate will be recognized throughout the EU.
Once you have these relevant stamps, the certificate will be recognized in the country / countries you intend to visit.
To sum up:
-> visa free travel is not possible because your wife's current permit is not as a family member of an EU/EEA citizen;
-> however, free and automatic visas are available from any Schengen country which you wish to visit, and once you get them they are valid in the entire EU;
-> for obtaining this visa, you will need a recognized marriage certificate, according to the procedure above.
I hope my answer was useful and look forward to your rating, which is essential to my activity.
Dr I L Vlad