The different post formats (quote, aside, etc.) are meant to be internal tags just as post and page are internal tags for post types and category and post_tag are internal tags for taxonomies. Internal tags are never translated, they are used internally to determine how various objects are to be handled. You likely realize this, I’m just clarifying what we are NOT talking about.
That said, the corresponding labels output by your theme for post formats should be translated. For example, the German WP still uses the taxonomy tag “category” internally, but all labels output related to the category taxonomy appear as “Kategorie”. Similar for post formats. The internal tag for quote format remains “quote”, but all labels output related to the post format appear as “Zitat”
Outputting proper translations of labels on the front end is the responsibility of your theme. If your theme is not properly translating format labels, it may not be well coded or support for your language has not yet been implemented for that theme. You could take up this issue with the theme author. A more expedient interim measure would be to determine why the labels are not translated in your theme and fix that yourself in a child theme.
I’m assuming the lack of translation is a front end issue and not in the admin area, since that part is the responsibility of your language version of WP. If you are seeing admin area labels that are not translated, the issue should be taken up with the package maintainers of that language version. If you need advice for reporting translation issues with localized WP versions, please ask on the #polyglots Slack channel.
