How can I get translation for a field of custom date
In Twig, you can use built-in translation contexts. I had to output my dates in dutch format. The default formatting was:
{% set date = node.created.value|date(&quot;j&quot;) ~ ' ' ~ node.created.value|date(&quot;F&quot;) ~ ' ' ~ node.created.value|date(&quot;Y&quot;) %}
// now {{ date }} outputs: 8 August 2020

When I use translation contexts, it worked as expected:
{% set date = node.created.value|date(&quot;j&quot;) ~ ' ' ~ node.created.value|date(&quot;F&quot;)|t({}, {'context' : 'Long month name'}) ~ ' ' ~ node.created.value|date(&quot;Y&quot;) %}
// now {{ date }} outputs: 8 augustus 2020

More about translation contexts in Twig: https://stefvanlooveren.me/blog/translate-dates-twig-drupal-8