"A pure number comes from combining three of nature’s most fundamental quantities: the speed of light, the electric charge carried by a single electron and Planck’s constant of quantum mechanics. In symbols, that’s c, e and h. Put them together as follows, hc/2πe2, and the units of c, e and h cancel out to leave a pure number, 137.03599913."
Because the units have cancelled out, the number is same in SI, British or any other unit of measurement.
Excellent read, article. 1/137.03599913 is called Alpha. And recently scientists discovered Alpha is slightly different in different regions of the Universe.
https://cosmosmagazine.com/mathematics/number-fascinates-physicists-above-all-others