All the discussions of the velocity of money that I've come across (e.g. here, here, and here) only display relative changes in the velocity of the money (e.g. by normalizing it by an unspecified value). I've never seen any source report an actual numerical value (with the units of inverse time) for a given point in time, for any choice of money supply. Does anyone have a reference for such a value?
(It appears that the answer was supplied in the original question. The third link was to the time series for the M2 velocity for the United States. This answer is just taking text out of the comments.)
The linked chart link to FRED website is of the ratio: (nominal GDP at an annual rate) divided by M2 (for the United States).
The units for annualised GDP are \$ per year, and the units for M2 are \$; the \$ cancels and we end up with 1/years as units. (This is tracking units the way that is done in the physical sciences.) If we took non-annualised quarterly GDP, the ratio would be 1/4 as large.
Some economics charts can be confusing from the perspective of physical sciences, because the inverse time units for flows are suppressed. This is a convention that has held for decades; you just have to know what is being done. It probably causes problems because people often look at mixes of stock and flow variables, such as the infamous debt-to-GDP ratio.