Yes MC14 and MC15 cannot coexist and so 14 is uninstalled during installation of 15 and as the 14 license is not valid for 15, its replaced by that trial license which is valid for 30 days only.
When you reinstalled MC14, how did you get your license? The license file usually is generated/sent during installation. If you have a single user license you usually have to provide the product code you got when buying MC and PTC checks if this code is still valid (the number of possible installations is limited) and then sends you a license file which is bound to your hardware.
In your case during re-installation of MC14 you should tell the installer that you already have a license and then point the license manager to a backup of the license file you got during the very first installation of MC14 (NOT to the license file you got when you installed the MC15 trial).
In case you did not make a backup of that license file and the number of allowed installations for your product code are already exceeded, you have to contact PTC support.
Its really bad and inexcusable that the trial version dos not make a backup of the old installation and offers no way back after the end of the trial period.