Per the OED, the word comes to us from the Old French assigner, past participle a(s)signé, meaning to assign. When Middle English adopted the word, it was spelled "assigne" and pronounced with three syllables. In the 15th century, the final 'e' on many words became mute and were eventually dropped. This happened to assigne, which became the modern assign. The same thing happened with avowe, which lost its final vowel to become avow.
To the victors go the spoils as well as the task of setting up the courts, which is what the Normans did in England after 1066. The language of the law courts was French and over the years developed into an archaic version of that language called "law French", which preserved the old vocabulary for legal terms. Presumably to preserve the distinction between the three-syllable law French assigne and the two-syllable, mute end vowel assigne, the former picked up an extra 'e' at the end.