Repost of solution as requested by erlend_sh:
Use Admin > Customize > Text Content …
search for 'YY which will list:
js.dates.tiny.date_year
MMM 'YY
js.dates.medium.date_year
MMM D, 'YY
js.dates.long_date_with_year
MMM D, 'YY LT
js.dates.long_date_with_year_without_time
MMM D, 'YY
js.dates.long_date_with_year_with_linebreak
MMM D, 'YY <br/>LT
Edit and adjust those replacing 'YY with YYYY to get the full year.