Same thing on this side…I was told that there would be a vertical stripe painted white to connect the stair trim to the baseboard trim but that seems off to me too since I feel like the trim should be connecting.

The above looks odd on both sides. Very messy + not very good planning. Is this a house you've recently purchased?

My staircase between the main level and the second floor is open on both sides = no walls = no stair stinger to meet any baseboard.

FYI - I have zero experience in carpentry - however, I did research in order to know how I wanted the below to look when completed.

I'm finishing my basement - this how the transitions from stair stringer to baseboards have been addressed (all will be painted white).

This how it looked before:

For your first photo, it would look better if the shoe moulding was removed from the top of the stair skirt and replaced with the top detail of your baseboard (you'd have to rip the top piece of a run of baseboard and add it on top of the skirting all the way up the stairs).

Look at the first photo that I posted above - the baseboard is made out of a flat piece with moulding on the top - then the same moulding is used along the skirt board.

You could add some type of wood plinth at the end of the corner of the wall + have the baseboard run into it (easiest).

I don't particularly like the pre-made plinth below - just showing an example of a plinth:

The above still looks better than what you currently have in your first and second photo.

For the second photo .- you could make your own plinth by using a piece of wood that is the needed height to meet the skirting - make it thinner to cover the end of the skirting so that it isn't thicker than the baseboard.

This photo isn't the same as how your stair skirt looks (yours runs to the end of the corner of the wall).

But if you use the above photo + the one below - maybe it will explain what I'm trying to describe.

You'd use a thinner board so that it would look better with the baseboard as was done in the above photo.

Accept it for what it is, unless you are willing to make major changes.

If, you want it to be flushed at the face. You need to get rid of the speed base and install a two piece baseboard. Your speed base is 1/2-5/8" thick. You need 3/4" thick material to tie into stringers. Also, you will need to replace the cap on the stringer so the profile matches the speed base. Then you would need to replace the rest of the house to match the new base board design.