What does it mean when a government agency insists that the translation of a document be “certified”? (J)

This simply means that the translator or translation company that carries out the work is required to append a statement to the end of the translated document stating that, in their view, the translation represents a complete and accurate rendition of the source document. The translator or translation company then stamps the statement with their official company seal.

Personal documents such as Japanese family registers, or birth and marriage certificates, normally require this certification step when they are translated for use in passport applications or other bureaucratic processes.

(Confusingly, embassies will sometimes post a helpful list of translators and translation companies on their website, with the caveat that the embassy makes no representation as to their competence.)

In any case, there is no official certification process (for translators or their work), so in practice this certification step is really just a formality that ticks a bureaucratic box.