Languages deal with possessive modifiers for family members very differently. Some languages use them much less than English, so word-for-word equivalences aren't inherently informative. Are you familiar with the situation in Swahili specifically? I share Miah803875's query as to whether "My dad..." might actually be closer to the default interpretation of this Swahili sentence.
Well, usually when you say "Dad is..." even in English the "my" is implied; since it is rather uncommon to refer to other people's fathers as "dad" or even "father" - I would say though that for Swahili "Baba ni mtanzania." should be "Father is Tanzanian." without "my" as that might stress biological father - Baba also is used for uncles and other elder men and not always specified (my uncle - my father's younger brother is "baba mdogo" to me - so my "small father")
Sure, "dad is..." means "my dad...," but to me this substitution isn't terribly natural outside extremely limited contexts, mostly talking to other family members. I wouldn't generally use it even when talking to close friends, and my friends don't seem to, either.
The simple fact they've included so many sentences with family members as subjects but without possessive modifiers makes me think the situation is different in Swahili. Of course, they could have just made a not-that-great pedagogical choice to teach lots of sentences that one would assume are natural and can be used in general, but actually are only really natural when speaking with one's Swahili-speaking family members, of which learners may often have none.
In English, "Father is Tanzanian" means only one of two things, as far as I'm aware: "my father is Tanzanian" or "the priest (and my interlocutor knows from context which priest I'm talking about) is Tanzanian." And "Dad is Tanzanian," or course, applies only to one's own biological father, so adding "my" doesn't add any stress.
To the extent "baba" is used for other men not one's father (barring priests), then "father" itself doesn't work as a translation, that is unless "father" itself is used in such instances in Tanzanian English. In Ghanaian English, it is common to use "uncle" in this way, a form much more familiar for not-actual-relatives to native English speakers in Western countries (but still not nearly as widely used as in Ghana).
Making "Dad is" become "Dad's" is quite informal in American english and is accetpable to say, however we are translating from Swahili and it makes the most sense to translate more directly into the closest equivalent. English-ifying it even more might be too far from the original meaning. The structure is much more like "Dad is" than "Dad's"