I understand that the literal translation of this sentence is 'my daughter wishes to have a horse', but I just think it is an awkward way to say it in English. I agree that wish is the best word for souhaite, absolutely. But, at least where I come from, wish has a sort of special grammar.
Sure, if you are blowing the candles off a birthday cake, or throwing a penny into a wishing well, I would say 'I wish for a horse' or 'I wish to have a horse'. But if I were using it in everyday speech (which is how this sounds in French), I would say something like, 'I wish I could have a horse'.
I realize that this gets into a whole different verb tense, but I think that conveys the same meaning as the French sentence better. Correct me if I'm wrong.
Apparently, the rules for using the verb "wish" in English are complicated. This is intuitive for English speakers and so many of the translations involving "wish" appear to be awkward. https://learningenglish.voanews.com/a/everyday-grammar-making-wishes/3218288.html
You are correct.
From other sources I was given to understand that souhaite referred to wish while espère referred to hope. Duo has reinforced this belief in their previous examples.
After reading your comment I looked at the drop down definition of souhaite at the top of this page and I see that Duo's current definition of souhaite is exclusively about hope.
Going to dictionaries provides definitions for souhaite that include both hope and wish. Espère is defined only as hope. Whatever distinctions French speakers may draw between the two words in everyday conversation, the literal, evidently grammatically correct use of hope in the English translation of this example should be accepted. Or, at least, until Duo itself stops defining souhaite as being about just hope and nothing else.
Part of the problem is that the distinction is being lost in English vernacular. For example, 'I hope that my sins will be forgiven through the blood of Jesus Christ.' That used to mean that I have a reasonable expectation that my hope will be fulfilled, because of God's faithfulness to fulfill His promises. But nowadays, that kind of hope has been relegated to the realm of myth and wishful thinking, along with the meaning of the word.
So, technically, hope is stronger than wish, and corresponds with espère rather than souhaite, but more often we use it like, 'I sure hope this works' when every reasonable indication tells us that it is a stupid idea.