Hopefully that's a piece of flat stock topped with a piece (possibly 2 stacked pieces) of shorter trim. If so, I would put a taller piece of flat stock on the shorter door, bringing the decorative pieces up to the same height as the top of the higher door. Then you can cope or miter the join.

It was probably done that way as the easiest method of joining the top sections due to the way they flare out at the top. Before you do anything, make models (possibly in clay or similar) of the two pieces to see what cuts you would have to make to join them in the corner. You may find it too difficult.

I would have put header trim on the closet so the moulding aligned.

What I see in old houses a lot is a corner detail like a corbel that means that each trim meets the corner piece, not each other at the corner. Additionally, if you have an old house, you're going to find lots of details like this. I think quirkiness is part of the fun of owning an old house.

If I put flat stock as a header, would you have the sides come all the way up, or leave the sides at the height they are and just dd additional header trim?

The moulded part is applied. Remove it on both doors. Remove the flat moulding off the closet. Reinstall the flat moulding on the closet so it aligns with the front door. Let’s say it’s 5 1/2”, replace it with 7 1/2” or whatever. Re apply the decorative moulding.