In a lighted fan, the fan MOTOR and the LIGHTS are actually two separate and independent devices in one unit. Each has its own cable to go to different mobo headers. So the LIGHTS in a fan have NO impact on the motor functions or its control.
punkncat above is right. Plain RGB uses FOUR pins in its header and supplies power to lights at 12 VDC. ARGB uses THREE pins (looks like it had 4 but #3 is missing) and supplies power at 5 VDC. Moreover its control signals are completely different from what plain RGB does, so you cannot mis-match the two types of lights and headers. If your mobo really has only plain RGB 4-pin header(s), you have two choices. The simplest it NOT to try to use the ARGB lights in any such fans or pump. The best alternative if you need the lights to work is to buy and install an ARGB Lighting Controller to feed those lights. There are several types of those, so if you need further advice post back here.
Other alternative is to replace the MSI Z390-A Pro mobo you say you have with one that has one or more ARGB headers. That may not appeal to you.
I've realized that Thermalright has many similar product names. I believe the cooler you are talking about must be the Phantom Spirit 120 SE ARGB. Right?