So, I installed a 73mm square taper bottom bracket into a 68mm shell. The drive side is fully in and good. But on the left side, it has a 2mm thread space outside. Is there a solution for this or simply let it go and it will not have any effect?

In order to install a 73mm bottom bracket in a 68mm shell, you need a 2.5mm spacer on the drive side. Without the spacer, the spindle will be off center

Or you could just install a 68mm BB in a 68mm BB shell.

You could source a spindle with the correct spacing for a 68mm.
That would be the only difference, unless this is a cartridge bb,
rusty

This^^^^

Wouldn't that move the chainline to the right 2.5mm?

Cheers

Yes, and that's the point. Installing the 73mm bottom bracket in a 68mm shell moves the chain line in by 2.5mm. The spacer puts it back where it should be.

As long as the drivetrain seems to work, I'd probably be more concerned about the security of the retainer cup on the non-drive side than an off-center spindle. In the days when 3-piece cranksets were standard, asymmetric spindles were fairly normal, and often deliberate to minimize q-factor (although this typically involved making the drive-side longer than the non-drive side, whereas the OP is doing the opposite).

If the chainline is problematic, it's possible that a 2.5mm spacer would "correct" the situation. Although given that the OP just shoved a 73mm bracket into a 68mm frame, I'd want to know more details before assuming that the spindle length is "correct" for the drivetrain in the first place.

Given that a Shimano square taper BB is about fifteen bucks, I can't really fathom how this is even a question.

$15 for a UN26. A UN55 starts at around $23.

A 2.5mm spacer is $4.25.

https://www.jensonusa.com/Shimano-UN55-Bottom-Bracket

Huh, that did not show up for me.

Still 3 times the cost of a spacer.

So a kludge rather than the correct part? I would rather spend the tiny bit extra and get the correct part. I could probably sell the 73mm BB and recoup some of the money I spent on the correct part. We aren't talking $100s here then I might say try a spacer but when you can do it correctly for not a whole ton of money and it will probably be a better more long lasting and reliable solution.

73mm BB in 68mm shell. Is there any legitimate reason someone would want to do this? Move the chainline maybe? Can't figure it....

A spacer is a kludge? So why do many high end BBs come with BB spacers in the box?

Because you have a 68mm bottom bracket shell and a 73mm bb on hand already?

It's happened to me.

My Phil Wood,SKF and IRD Defiant bottom brackets didn't come with spacers. I just ordered the correct sizes for the correct shell width. Of course I am talking square taper as you are talking probably external BBs which do sometimes come with spacers but I have never needed to use those, sometimes for the cranks yes but never on the actual threaded part.

OK, well the OP now has choices. He can go to his LBS today and pick up a spacer for $5 or he can order a new BB from Jenson for $22 shipped and wait 5 days for it to accomplish the same thing.

I know what I would do, but that's just me.

FWIW, I've run into the exact same situation as the OP (had a 73mm ST BB and a 68mm shell, and I used a spacer. Worked fine for years until I got rid of the frame.