This is a subtle mistake of the formatting string, which is that there are multiple different year specifiers:

• A lower-case yy or yyyy is the normal year you would expect, for use with month and day
• An upper-case YY or YYYY is the year for use with "week of year"

For the majority of dates, these are the same, so the difference is easy to overlook. But at the very beginning and end of the year, it matters where the week begins and ends - January 1st may still be in the last week of the previous year, or December 31st may be in the first week of the next year.

We can see the difference clearly if we use a format string of 'ww YYYY, MMMM yyyy', to show the week number and both definitions of year:

• For en_GB we get "01 2022, January 2022"
• For fr_FR we get "52 2021, janvier 2022"

So always use the lower-case "yy" form, unless you are actually displaying week numbers.

function formatted_month($month) {
    $first_day_in_month = new DateTime('2022-01-01');

    $fmt = datefmt_create(
        'fr_FR',
        IntlDateFormatter::FULL,
        IntlDateFormatter::FULL,
        'Europe/Paris',
        IntlDateFormatter::GREGORIAN,
        'MMMM yy'
    );

    $formatted_month = datefmt_format($fmt,$first_day_in_month);

    return $formatted_month;
}