https://www.thenation.com/article/society/shlomo-avineri-karl-marx/
Marx is both the father of what we call nowadays social-democracy and communism. However, the social-democrats in the West rejected most of Marx during the end of the 19th Century, and here I'm specifically talking about the SPD (German Social-Democratic Party), which was, as the old saying goes, "the party not of Marx, but of Lassalle". The SPD was to the social-democratic parties what the Russian Communist Party (future CPSU) was to the communist parties, i.e. their leadership.
In the UK, Labour was not founded and never was a Marxist party, but a pure trade-unionist party. It was "retconned" as a social-democratic party only after WWII, when the USSR emerged as a prestigious world superpower and where a legitimizing narrative was necessary to justify the Attlee reforms of 1945-1950. But, in general, social-democracy and Marxism were never really a thing in the UK, save for some very marginal, underground, groups.
Marx's loss of the SPD to the "Lassallists" generated one of his most famous books, entitled "Critique of the Gotha Program". After that, he essentially gave up trying to take the party to his side. The SPD continue to pay homage to Marxism until Engels' death (1895). After Engels' death, the SPD cut all ties it had with Marxism and became a Lassallian chimera with the help of Kautsky and co.
The definitive schism of social-democracy - that would give birth to the division we know today between social-democrats ("socialists") and socialists ("communists") - would finally happen in 1919, with the foundation of the Third International.
So, to sum it up, not only it is an anachronism to call Karl Marx a "social-democrat", but, if such revisionism was to happen, it would be the communists - and not the social-democrats - who would have the better claim to the sole possession of Marx's affiliation.
P.S.: Marx wasn't a Jew. His father was a Jew. You're only born a Jew if your mother is a Jew, not your father. But even his father was a secular Jew, and immediately converted to Protestantism in order to take an office in the Prussian State. Marx didn't receive any Jewish education, and there is no evidence he ever practiced Judaism, nor frequented Jewish circles. Either way, being a Jew doesn't indicate at all that you're more naturally inclined to social-democracy than to communism: the Bolshevik Party was filled with Jews (Trotsky was a Jew). The "Bund" (the Jewish faction of the Russian Social-Democratic Party) was neither affiliated to the Mensheviks nor to the Bolsheviks. The myth that the communists hate the Jews probably comes from the post-war phenomenon of the so-called "Refuseniks", and from the stereotype in the West that the Jews are impassioned capitalists.
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