2 Apple IDs/Family sharing

My original Apple ID was used for iTunes so long ago that it had a user id associated with it rather than an email.

When I got my first iPhone, it wanted an Apple ID with an email address, so I created a new one instead of realizing I could update the original account.

So I have one account for purchases (the user ID has been changed to an email address) and one for everything else.

I have Family Sharing enabled.

My question is...I receive periodic messages on my Mac that I don't receive on my iPhone. I'd like to get them on the phone so I can respond to them timely. Am I misunderstanding that I should be able to receive both on the phone because of Family Sharing?

The phone is signed into iMessage with the "new" account

The Mac is signed into Messages with the "old" account

I'd appreciate any help!!