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I don't know about other EU countries, but everyone in Germany has a certificate of vaccination, a yellow booklet 12 x 9 mm which contains all the person's vaccinations: 'flu. shingles, brucellosis (from ticks) etc. It seems an excellent idea & your doctor can see when it is time for renewals & boosters.
It will now work well for covid vaccine receivers (when they get them!) to allow freer movement, I can't see why anyone would object to its implementation.
Could it be the added cost to the NHS, I wonder ?