Why do we need to measure the vibration of rotating machinery (e.g., acceleration, velocity, and angular velocity) in three axis?
I am wondering Why we need to measure the vibration of rotating machinery (e.g., acceleration, velocity, and angular velocity) on the three-axis? why single-axis vibration signals (e.g., only X or only Y, or only Z) are not enough?

To research what the machinery does initially, you need to collect all the information you can. You don't know what it's going to be doing, maybe the key signal you want is going to be on the X, Y or Z axis, so look at them all.
Once you've characterised what the machinery does, you might find that just two or even one axes are sufficient to extract the information you want.

Also, please, is there any way to integrate/combine the 3-axis signals into a representative single-axis measures/metric. For example, combining the acceleration signals on X, Y, and Z into one representative signal!

There are several methods of reducing the number of axes you have available. For instance, the rms of the three vectors would give you something like the total energy in the vibration. However, this might not be what you want, if the machinery has a normally operating strong X axis vibration, and the fault-indication vibration is a tiny Y axis acceleration. You would need to try out several methods in your initial research phase, and see which one worked best for your particular device.