Hubby wants to open a savings account. He said it's for us. But wants it to be under his name. Does he makes sense? I want it to be under both our names though so whatever happens I know I also have authority and for sure I'll contribute to it in the long run. What to do?

If both of you are contributing why not put both names? Absolutely not to put only under his name especially if he says it is for both of you. With a joint account it will be easier to withdraw money in case something happens to one of you. And make sure you check the transactions every month