Hey Jace,
these milling motors made by Mafell and AMB (former Kress) are, like the Makita hand trim router, double-insulated Class II (IEC 61140) devices that are not grounded. However, the Onefinity CNC is a grounded Class I device. All of its components that are able to carry current must be grounded. 
Important: The electrical connection of the individual components of the machine via the linear bearings or the ball screws is not a valid protective ground connection. Each moving part must be connected to ground with its own protective earth conductor. As mentioned in the thread linked above, grounding the entire CNC machine in accordance with safety regulations is not only mandatory as of the safety standards, but also noticeable in terms of preventing EMI and eliminating electric shocks due to potential differences.
Note that the use of a TN−C earthing system with PEN conductor (combined PE and N conductor) breaks the protection concept of protection class I.
The additional importance of proper grounding when using a spindle is that it is earthed and that it is driven by a VFD that uses high carrier frequencies internally. Proper grounding, shielding and filtering is necessary to prevent motor bearing currents which can damage the bearings of the spindle.
By the way, switching from the router to a spindle is also the reason why then you don’t need the magnet end of Touch probe anymore. Since the spindle is earthed, the Touch probe connection only needs one wire (see here).