Hi @wslizgiem!
Thanks for choosing Neve!
Unfortunately we were not able to replicate this issue, the Customizer works as expected on our test instances. Does this still happen if you deactivate the child theme and use only the main one? When you update the theme, only the parent one updates, not the child theme. This means that in the child theme you will still have the files from the previous version, not the updated one. Also, the changes made previously within the child theme (and the old files they were made in) still override the updated parent theme and this could cause the issue. You could try to backup your site and remove one by one all the files that exist in the child theme. This way you will be able to check which file could cause the issue, replace it with the same file from the parent theme (so the file will be updated) and then add again the changes that you made in the old file.
I hope this helps!
Have a nice day!