The RPi5 has a different DSI connector to the RPi4 hence you need a different cable. The link you posted is for a screen compatible with RPi3 & RPi4. It does not mention RPi5 and the cables in the picture look the wrong type. There are displays that provide both types of cable. I bought the Waveshare 5inch which is more expensive and has some issues but mostly works. The display described by @stojos, the architect of the zynthian mini is probably the best option but does require the additional cable for the RPi5.
The cable is fairly cheap. I bought a longer one so that I can mount my screen more flexibly but the short one works well if attaching the Pi directly to the back of the screen.
The problem I have with the waveshare is that touching it whilst the driver is not loaded breaks touch, e.g. touching the screen whilst powering on. Maybe someone with another similar screen (like @stojos) could test if the same behaviour exists.