Why 'prefecture' for Japanese administration areas?

Solution 1:

From Wikipedia--

The West's use of "prefecture" to label these Japanese regions stems from 16th-century Portuguese explorers' and traders' use of "prefeitura" to describe the fiefdoms they encountered there. Its original sense in Portuguese, however, was closer to "municipality" than "province". (Today, in turn, Japan uses its word ken (県), meaning "prefecture", to identify Portuguese districts while in Brazil the word "Prefeitura" is used to refer to a City Hall.) Those fiefs were headed by a local warlord or family. Though the fiefs have long since been dismantled, merged, and reorganized multiple times, and been granted legislative governance and oversight, the rough translation stuck.

Solution 2:

Préfecture is the French word used to describe small administrative areas (i.e. "départements"). The French administrative model inspired Japan when reorganizing its administration.

From French Wikipedia préfectures du Japon:

Cette organisation a été établie officiellement par le gouvernement de Meiji en 1871 dans le cadre d'une réforme dite « abolition du système han » sur le modèle français. Il a remplacé dans les faits les anciennes provinces, bien que celles-ci n'aient jamais été officiellement abolies.

Translation: this new administrative organisation was officially established by the Meiji government in 1871 as part of a reform called Abolition of the Han system based on the French model. It replaced the former regions, even if they were not officially abolished.

In France, the word "Préfecture" has the following meanings:

• the position of prefect,
• the city where the prefecture is located, i.e. the capital city of the administrative entity("département"),
• the administrative services supporting the prefect activities,
• the building where he lives and where are located his services.

From on-line etymology, the English word "prefecture" is:

an administrative district of a prefect, mid-15c., from Middle French préfecture and directly from Latin praefectura, or assembled locally from prefect.

  EDIT #1 to respond to @HotLicks comment:

After 2 centuries of isolationism during the Edo period, the Meiji government conducted different reforms, most of them being done after studies of what was done abroad. For example, one of the Meiji oligarchy, Itō Hirobumi (1841–1909), a Chōshū native long involved in government affairs, was charged with drafting Japan's constitution. He led a Constitutional Study Mission abroad in 1882, spending most of his time in Germany. He rejected the United States Constitution as "too liberal" and the British system as too unwieldy and having a parliament with too much control over the monarchy; the French and Spanish models were rejected as tending toward despotism. source

  EDIT #2 about French influence on the new administrative organization:

France had been using the word préfecture since Revolution and Napoleon’s time for its own major civil administrative units. Napoleon was seen by the Japanese ambassadorial legations as a revolutionary emperor who modernized and expanded France.

Note that using the term "préfecture" to refer to Japanese administrative is an "anglicism" because, in French, préfecture is not used to refer to the area governed by the prefect. Purists prefer "département" to translate in French the japanese term "todōfuken".