Translation:My brother does not shower every day.
Can be interpreted either way (in spoken speech) and, if someone wants, start to joke with the whole situation (specially if emphasis is given to "todos los días").
In any case, the usual interpretation is that the other person showers every other day (but not daily).
Se duchar/ Ducharse: Showering oneself, Me ducho : I shower (myself), Te duchas: You shower (y.s.), Se ducha: (S)he/it showers (her-/him-/itself), Nos duchamos: We shower (o.s.), Vos ducháis: You (pl.) shower (y.s.), Se duchan: They shower (t.s.) (Where the -n indicates plural)
"se" is a reflexive pronoun (either in the form of "se ducha" or "ducharse") and identifies to whom the action will be performed... Check the "Reflexive Pronoun Forms and Examples" in https://www.spanishdict.com/guide/reflexive-verbs-and-reflexive-pronouns
Another option can be due to the recent course reorganization... for instance, I have many lessons completed near the end of the Spanish course but the last reformulation created a whole bunch of lessons in the middle of the course that need to be completed (this is one of these lessons).
My brother doesn't have a shower every day.
It was accepted because 'have' is a 'delexical verb' and 'have a shower' is a 'delexical structure'.
I guess it is not usual to explicitly include the reflexive part in English... which to me is not a reason for considering a sentence incorrect.
Note: as seen in other discussions, it is very likely that someone will state that you did a literal/word-by-word translation.