Apple made two versions of the 27" display.
The older one known as the LED Cinema Display (27-inch):
LED Cinema Display (27-inch) - Technical Specifications
"One cable with three connectors: Mini DisplayPort with audio support, Universal MagSafe (up to 85W), USB 2.0"
To connect this to a new Mac mini you need a USB-C to Mini DisplayPort adapter, such as:
https://www.startech.com/en-us/display-video-adapters/cdp2mdpec
The newer one known as the Thunderbolt display:
Apple Thunderbolt Display - Technical Specifications
One cable with two connectors: Thunderbolt 2, Universal MagSafe (up to 85W),
To connect this to a new Mac mini you need a Thunderbolt 3 to Thunderbolt 2 adapter:
Thunderbolt 3 (USB-C) to Thunderbolt 2 Adapter - Apple
The adapters are not interchangeable, even though the plugs will fit.