This is a good question, because Franco's obsession with Freemasonry is just so damn weird. Prejudice against Freemasons was hardly unknown in other fascist or quasi-fascist regimes of the mid-twentieth century, but no one else showed nearly as much interest or dedicated in rooting out the 'threat' they posed. Even the Nazis, while identifying Freemasons as a potential enemy category on their seizure of power, had largely lost interest in them by the end of the 1930s. By the outbreak of war in 1939, while there was in theory a 'Masonic' desk of the Reichssicherheitshauptamt (Reich Main Security Office), it had been left unfilled. That's not to say that no Freemasons were persecuted by the Nazi regime, but they simply were not a major priority.

In Spain, in contrast, Freemasonry was always a prominent Francoist enemy. Not only were Freemasons among those sought out during the civil war itself, wholesale postwar repression was codified in the Law for the Repression of Freemasonry and Communism of March 1940 - it's telling that Freemasonry was given equal billing with Communism in this key piece of legislation. The desire to purge masons from all levels of the new Francoist state meant that even Freemasons who had actively fought for the military rebellion were purged from the army and civil service once their masonic connections came to light. Even those who sought to hide their former affiliations tended to be discovered, thanks to the repressive apparatus created by the new laws, with Republican and other documents captured during the civil war used to create a massive archive in Salamanca that was rigorously employed to ensure that potential enemies of the regime were rooted out. This hardline, paranoid concern with Freemasons was to last for the duration of the regime, and was by all accounts shared by many of Franco's closest associates such as Luis Carrero Blanco.

We're still left with the question of why. Here, we need to acknowledge that suspicion of Freemasonry has a long tradition in a Spanish context. Freemasons, for instance, were among those blamed for the loss of the Spanish-American war of 1898. This in turn built on decades of papal attacks on Freemasonry as an international conspiracy seeking to overthrow the Christian political order, which found ready ears in Spain, as the subversive actions of a shadowy global conspiracy offered an appealing explanation for the decline of Spanish imperial power over the past century. As calls for secularisation of Spanish society grew in the early twentieth century, Freemasonry remained a natural scapegoat for opponents. The relationship between Freemasonry and these political movements was not entirely imagined - the new Republican deputies elected in 1931 after the overthrow of the monarchy contained disproportionately large numbers of Freemasons, reflecting not only the movement's importance as a home for secular, liberal thought, but also the small, elite circles it tended to recruit from.

For Spanish traditionalists - for whom the special relationship between church and state was the foundation of Spanish identity and historical imperial success - Freemasonry's association with liberal secularism was therefore a direct threat, an ideological cancer that threatened to undermine what made Spain great. For Spaniards like Franco, re-establishing the true Spanish state meant defeating 'foreign' ideological enemies that had supposedly undermined it from within. Here, an interesting comparison with Vichy France might be drawn, a state similarly concerned with the restoration of traditional Catholic values and governance, and expunging the weakness caused by secularism. Unlike Vichy, of course, Franco had considerably more opportunity to follow through with his vision. The civil war, which after all had been won fighting against 'anti-Spanish' elements within the country, afforded Franco the opportunity to take the fight to these imagined enemies within.

The categorisation of Freemasonry as a foreign ideological conspiracy to destroy Spain did, as you suggest, intersect with other forms of conspiratorial thinking steeped in anti-Semitism. Freemasonry, in this analysis, was akin to communism - an international(ist) conspiracy orchestrated by Jews to destroy traditional Christian society. Just as the Communist International had been organised by Jews, Freemasonry was an even longer-standing Jewish plot to destroy Christian civilisation. While the Franco regime was considerably more pragmatic in its anti-Semitism than Nazi Germany, even going so far as to bow to pressure to allow some Jewish refugees across its borders (on the strict condition, of course, that they did not stay in Spain indefinitely), there is no doubt that these ways of thinking were deeply entangled in Franco's and other regime figures' world views.

The persistence of the anti-Freemasonry campaigns also owed a great deal to the their failure. While thousands of Freemasons were investigated by tribunals, little direct evidence of a grand masonic plot emerged. Rather than evidence that perhaps Spanish Freemasonry was a small, fractured movement with limited influence even over its own members, this was taken as evidence that the wily Freemasons had slipped through the Francoists' fingers, escaping with their records into exile, leaving behind only the small fry who had little idea of the movement's true purpose. This, of course, justified further paranoia and redoubled efforts to combat masonic penetration, especially in the armed forces, where it was feared that masonic subversion could do the most damage. Even by the final years of the regime, Freemasonry was still being held up as an existential external threat to the Spanish state and traditional way of life.