Since 2014, Twitter makes it possible to send a tweet with pics that have Twitter users tagged in them (ref).
How can I do that through the Twitter API, and with Twitter4j if possible?
It looks like they have released this as a feature of their App but have not released any API's for the same till now.
You can check following forum which mentions Photo Tagging API is not publicly available yet. As per official update from twitter staff on 17th April, this is not made public yet. You can follow @twitterapi / @TwitterDev to get updates about new API's they add.