The idea that these are the “only two classes that really can’t help taking life seriously” is (like everything else Reginald says) utter nonsense, but is there a joke or satirical point behind it? I have a theory, which is Reginald is implying that these two classes of people are similar in another way.
When Reginald was published in 1904, the head of the House of Hohenzollern was Wilhelm II, emperor of Germany, who was:
superficial, hasty, restless, unable to relax, without any deeper level of seriousness, without any desire for hard work or drive to see things through to the end, without any sense of sobriety, for balance and boundaries, or even for reality and real problems, uncontrollable and scarcely capable of learning from experience, desperate for applause and success,—as Bismarck said early on in his life, he wanted every day to be his birthday
Thomas Nipperdey (1992). Deutsche Geschichte 1800–1918, volume III, Machtstaat vor der Demokratie, p. 421. Translated by Richard Evans (1997). Rereading German History: From Unification to Reunification, 1800–1996, p. 39. London: Routledge.
Based on Nipperdey’s description of Wilhelm’s character, I think Saki’s joke may be that the German emperor resembles a stereotypical schoolgirl.