"Depuis que tu as un chapeau."
Usually, when you learn a new expression that has a single meaning, like "parce que", the entire expression is highlighted, indicating that both words combine to have that single meaning. In this case, only "depuis" is highlighted, and so, the additional "que" becomes confusing. It seems to imply that "que" has additional meaning, when, in fact, it does not. You should highlight both words to maintain consistency, unless this is some sort of a ploy to get the user to realize that "que" is often tacked onto other words to create expressions.
I think that they are trying to emphasize the use of "depuis que" along with the present tense to indicate a sort of past-tense connotation. For example, in this case, an acceptable translation (from what I have seen) would be "Since you have had a hat". There is another example where it is translated as "She has been writing since she was young." (Elle écrit depuis qu'elle est jeune). But I'm not a French expert; this is simply what I've divined from the exercises. So yes, essentially, "Since she got a hat" would have the same meaning, but it's a bit more informal.
The example you've given would be "Puisque": "Puisque tu as un chapeau, tu ne vas pas se bronzer en marchant." The more I see Duolingo translating "Depuis" into English with the present tense like this, the more I'm reminded how wrong this is. Because in English we only have one since to cover two meanings (temporal and causal), we use the tense to distinguish between the two. For example "Since you have a hat, you are not getting sunburnt." is causal (and would translate with present tense + puisque). "Since you have had a hat, you have not been sunburnt. But not because the hat is a sun hut: it's a woolly hat, and it's Winter, and that's why you aren't getting sunburnt." is temporal (and would translate with present tense + depuis).
I wrote "Since you have a hat" and was marked correct but after reading this discussion I think Duolingo should actually tell me that answer is wrong? Sounds like the meaning of "depuis" is "since" in the time sense, not the causative sense. So "Since you had a hat" or "Since you have had a hat" would be an ok translation, but "Since you have a hat" is wrong?
From reading other duolingo discussions I think it's temporal. Hence my other comment. I think in English it needs to be "have had a hat" or "had a hat", not "have a hat" as that implies causality and should be translated using "parce que". But I'd really like a French speaker to confirm that this is right!
Accepted spelling errors are a bonus. For the life of me I can't figure out how they program spelling latitude into the computer. Maybe consonant errors are acceptable but not vowels, or maybe the other way round. Maybe every third one is ok.
I'm surprised that spelling mistakes are ever allowed in the French parts.