We do not have such a sentence in the course. If we did, I would translate it as "My daughter is dirty.".
However, you certainly may be just commenting o photograph of a family and say "The daughter is dirty.", yes, that is possible. A demonstrative "ta dcera" may be in order if it is an isolated comment, but would not be present in a longer description with all the persons.
But I am not so sure that could be done with the sentence we have here.
Is "The grandmother ... " real grammar mistake? Or it just sounds strange to you so its marked as a mistake? (and i do not mean it wrong, I'm just fairly asking)
It would help if you allow more options if they are grammatically correct. (or at least they are not grammatically incorrect :-) )
That is a different course. I can speak for the English course for Czech speakers. I do not contribute to this course.
I also do not feel confident to teach you English here, this is a course of Czech. Native English speakers may be able to help you better.
It is not a complete grammar mistake, the sentence is grammatical, but I am not sure if it makes good sense.
I am native AmE.
As VladaFu says, the sentence with "the" (or with "a") is grammatically correct. But grammatically correct and "usable" are not always the same thing. "Airplane files up" and "Tall girl is there" are, strictly speaking, grammatically correct, but they would not be used in conversation or in writing. But more importantly...
With relatives and body parts, if there is no possessive pronoun or demonstrative in the Czech sentence, it is generally understood that the relative or body part -- here, a grandmother -- is "owned" by the speaker. That is one of the things this exercise teaches. Both "Grandmother" (used as a name) and "my grandmother" are accepted. (Edited for clarity, 5 Feb 2020)