while उसे means "to him" or "to her", उसके means "his" or "her"
उसके is the plural masculine version of उसका
उसे नाश्ता चाहिए can roughly be translated as "by him breakfast is wanted". But in English it is of course more convenient to phrase it as "he wants breakfast"
Another example is उसे सलाद पसंद नहीं है where the literal translation is "to him salad is not pleasing". However, we would phrase that as "he does not like salad". Don't let that confuse you, there is a reason for why उसे is used in hindi even tho it might not seem obvious at first!
Examples where उसके is used are:
उसके बेटे भारत में हैं = "his sons are in india"
and उसके कुत्ते बड़े हैं = "his dogs are big"
Because when चाहिए is used, the subject (in this case "he") always needs to take the dative case.
Think of this sentence as "By him breakfast is wanted". If you used वह it would roughly translate to "he breakfast is wanted" and even though the meaning can be interpreted, it is not gramatically correct.
The same way "I want breakfast" would be translated as मुझे नाश्ता चाहिए. That is, मुझे is used instead of मैं.
See this link for more info: https://www.duolingo.com/skill/hi/Food/tips-and-notes