Since they are both just phrases, it is more difficult to understand the meaning. But as a child we had a rhyme:' First star I see tonight, I wish I may, I wish I might have the wish I wish tonight." If you know this rhyme, it helps explain that we can be wishing on a star, for a date with someone you like, for good weather, to pass a test, things like that. This structure is more usual and whimsical in meaning. Using "wishing for someone" might mean wishing for love, or it could be incomplete and mean I am wishing for someone to get a new job, to get a new house, or to win the lottery, but I think we would be more likely to say "I wish he or she would get a new job or win the lottery.

Hi :) I'm wishing on a star =Usual. I'm wishing for someone. =Unusual (We would say instead "I'm praying for someone right now" :)