Thank you to @Layn M for posting your comment as this has worked for me at last! This has been bugging me for YEARS! But it works even with mm or any measurement that you choose (not just Pixels).
The trick is (like you said) - AFTER YOU CREATE YOUR PRESET - that was the step I was missing! So you open InDesign and before you open any documents, you edit the Units and Increments preset from the Preferences menu item.
Then you following the File > Document Presets > Define steps and you should see next time you close and open Indesign it will remember your measurement.

Oh what a day!

I had to share this because I was having the same problem, but I managed a fix if you are going for Pixels.

After you've created your preset, go to
File -> Document Presets -> Define..
Click on your preset and press 'Edit..'
Select the 'Intent' drop-down and select 'Web'

Your preset will now always load as Pixels and not Picas.

This is frustrating - the whole point of presets is that once you go to the effort to set them up, they work out of the box. I'm having this same issue now.

well its 2022 now and.... still nothing - awesome work. I HAVE THE SAME PROBLEM even in the most recent update. Are you people tone deaf?

This has been a problem since 2017! And this issue has yet to be resolved in July of 2021. Very poor development on Adobe's part.

Same problem. Ridiculous. Please fix ASAP.

Having the same issues with ID 2021.

Another person here with this problem. It's since updating to InDesign 2020.
I have to change my preference to mm every time I fire up the computer (which is everyday)
It also doesn't keep my saved templates.
I've tried deleting preference files and all of the things on the internet, with no luck

On InDesign 2021 and all presets are shown in picas..... I don't even know who uses that measurement...... PLEASE FIX!

Same problem since updating to InDesign 2020.
Changing units or creating new presets doesn't work. Very annoying!

is there a solution to this problem?

I have this problem as well.

I have same problem. Units & Increments reverts back to picas no matter what other measurement is chosen.

i have the same problem.

In the preferences I change the units to inches and it saves. But when I open and close the program again, InDesign, Photoshop, etc., the units reverse back to picas