What is known about the Sage of this name?
According to Rabbi Reuven Margaliyot in "Lecheker Shemot V'Kinuyim BaTalmud", pg. 12, "Munbaz" was the nickname of this individual, called so by the sages because of his particular interest in questions about converts and tinokout sh'nishbu (captive children), because King Munbaz was a famous convert (see @RenatoGrun's answer). He (the tanna) was a student of Rabbi Akiva.
It occurs to me, though, that as there were Jews in Chadayev, the kingdom from where King Munbaz came from, it's possible that this Munbaz (the tanna) came from Chadayev (although that would make him particularly unusual, because it seems that in Chadayev there wasn't strong mitzvah observance and adherence to Chazalic laws), and that either Munbaz really was his name or that he received his nickname because of the famous convert from his country, but this is just my opinion.
It should be noted, however, that Rabbi Mordechai Hakohen in his book בעיני חז"ל (In The Eyes of The Sages), pg. 165-166, brings sources from the Geonim and Rishonim who had a tradition that Munbaz the tanna's name was with a Nun, while king Munbaz was actually called Mulbaz, with a Lamed, but eventually these traditions were lost and only the name Munbaz - for both figures - remained. One such source is the Aruch on Malbaz:
"מלבז [נאמע] (יומא לד) מונבז המלך עשה כל ידות הכלים פי' מולבז המלך בלע"ז מינבז תנא בגמ'"
Translation: Malbaz [a name] Munbaz the King made all of the handles of the vessels, meaning, Mulbaz the King in foreign languages, Minbaz the tanna in the gemara.
Rabbi Hakohen explains per Kohut's annotated edition of the Aruch that the Aruch should be read as such:
"מלבז - מונבז המלך עשה כל ידות הכלים פי', מולבז המלך בל' מונבז תנא בנ'"
Translation: Malbaz - Munbaz the King made all of the handles of the vessels, meaning, Mulbaz the King with a Lamed, Munbaz the tanna with a Nun."
Per such an understanding, there may not be any connection between Munbaz the tanna and Munbaz (Mulbaz?) the King.