The requirement for a special (ink) stamp for use on Money Orders was included in the instructions for postmasters at all Money Order Offices when the Money Order System began operations November 1, 1864. Here is the instruction pertaining to the special stamp, included in the "Instructions to Postmasters of Money Order Offices", printed in the November, 1864 (Vol. V, No. 2, Whole Number 50) United States Mail and Post Office Assistant.

Note that it explicitly states this special stamp
shall never be used for stamping letters.
Initially, the postmasters procured these date-stamping devices. Given this latitude, many different shapes, sizes and styles were purchased, depending on each postmaster's preference. According to the 1887 Postal Laws and Regulations, the Post Office Department was providing the "M.O.B." date stamps to the Money Order Offices. This was likely an effort to standardize the design used by all of the Money Order Offices. In 1887, the newly formed Postal Note Offices were also issued standardized "P.N.B." date stamps.
So anytime you see a "M.O.B." or "P.N.B." date stamp on a cover, that use was actually in violation of the Postal Laws and Regulations. Examples seen were probably stamped in error (they grabbed the wrong stamping tool) or out of necessity (the postal stamping tool was worn out.)