I just can't understand why Howard was against Joanie having her own car at 16 when he had helped Richie to get one at the same age. Was it because she was a girl?

I think this is a really good question. Maybe he was just more protective of her as his daughter as you've indicated.

Another thing worth mentioning is that I'm guessing it was somewhat uncommon for young women to have cars in the timeframe the series harkened back to. Either way, apparently and simply her being a girl seems to be the answer.