Chosen Solution

Is it possible to put my iPhone 5s motherboard into a iPhone 6 or 6+? Even with soldering? I know the cables and parts don’t fit but is there a way to make it work?

No. Practically impossible as you will run into many issues which outweigh the reason of even doing so, it will just result in time and money wasted or lost. It is theoretically / technically possible if you had the time and lots money of money to spend. TL:DR - No.

No, nothing fits. This is nothing to do with soldering.

You can buy a backhousing for the 5S that will make it look like a iphone 6 on chinees website.