The approach I would use would be to remove the drive from the old enclosure and either put it in a 2.5" USB enclosure or get a USB-to-SATA adapter. Then plug that in to the new Mini and verify that the drive is operational and has all your data.
Next, I would proceed to do a new clean install on the new Mini. I boot the new Mini in its current state and log in and then create a new MacOS USB installer on a USB Flash drive with the version of MacOS desired (note: it should be at least the version that you were running on the old Mini or newer):
Create a bootable installer for macOS - Apple Support
Once complete, boot to the USB installer (hold down the option key when rebooting and select the installer). Once the installer loads, select Utilities->DiskUtiliy in the menus. Use it to reformat the internal drive. When complete, go back and start the installer. Let it do its thing and after it installs and reboots it will come to a point where it will ask if you want to migrate from another computer or Time Machine backup. Plug in the old drive to USB and continue following the on screen instructions. At a point you will be prompted with a screen giving you the option of what to migrate. Select all the items you wish to migrate and click continue. This migration part may take some time depending on how big your old drive is and how much is on it. when it is complete, you should have it working nearly exactly as your old Mini.
I strongly suggest this approach as opposed to swapping drives as you don't need to open up the new Mini and possibly damage something in the process.