"Rehema is Tanzanian."
Translation:Rehema ni Mtanzania.
The m-prefix is not posessive but indicates the noun-class (in this case it's the class for people, human beings, etc.) and the tempus. In this class, m- indicates singular, wa- indicates plural. There are a lot of different classes for different categories of things in Swahili. The posession is not indicated in the noun itself but through the posessive pronoun. This sentence in particular has no posessive form, by the way.
-ake is specifically for "his" or "her". With a following noun, it's just -a, which is roughly equivalent to "of".
Jina lake = her/his name
Jina la tembo = the name of the elephant
The prefix on the possessive word, in this case l-, depends on the noun class of the noun before it.