The middle version of the tag is the older ASME Y14.5 version (ANSI Y14.5). The newer one on the right is an ISO allowance that has become the staple in most drawing standards today.
You can re-assign datum tags to surfaces. However, there will still be a datum feature such as a plane or axis as the identifying reference.
I will say this; do not use any of the primary datum planes to assign datum tags. Use a surface to define the datum "intent" surface and let the system make the plane to create the tag. It is all very convoluted, but again, the tag belongs to either a datum plane or a datum axis. Re-assigning it later to a surface or a dimension can be done but the plane and axis will remain.