This is an interesting question that links to many other debates about the nature of fascism and of Francoist Spain. There have been a tonne of fantastic and comprehensive answers about fascism in this subreddit, so you may find seeking those out to be helpful for further reading.

Your question is in two parts though: 1. was Franco 'ideologically similar' to Hitler & Mussolini. 2. Did Franco keep Spain 'neutral' in the war, if so, why?

Here is a comment I wrote adressing the Fascist elements of Franco's regime. What that comment mentions is that there was a very Fascist element to Franco's regime, the 'Falange' (who became the sole political party), however the Falange weren't fond of Franco and likewise I mention a couple of historians that argue Franco was actively trying to remove the fascism from his regime "as early as 1942". A group in the Falange even tried to have Franco assassinated in 1940.[1]

The argument put forward there is that Franco 'played' the Falange; he gave them a great deal of power and a central place in the Spanish dictatorship, but similarly hollowed them out and filled their ranks with military men who personally backed him. Paxton argues Franco "subjected Spain's fascist party to his personal control",[2] and similarly Payne describes the Francoist regime as one in which power was unusually concentrated with Franco himself (which even the Nazis and Italian Fascists did not have).[3]

The fact remains, however, that Franco did embrace the Falange into his regime, and certainly acted in many ways that were rather Fascist. That particular list would be rather long, however, so this quote from Paxton serves as a concise yet fair summary, in my opinion:

"Virulently hostile to democracy, liberalism, secularism, Marxism, and especially Freemasonry, Franco joined Hitler and Mussolini in April 1939 as a signatory of the Anti-Comintern Pact" [4]

Indeed, it is well documented that Franco received assistance from the Axis powers during the Spanish Civil War, especially from Italy[5] (although the Nazis carried out the infamous bombing of Guernica). Payne writes:

“Though Franco never fully adopted the entire core fascist revolutionary ideology, there is no question that he identified his regime politically with the fascist powers and considered himself not merely an associate but virtually an ally of the Axis.”

"In the closing days of the Civil War, Franco signed a treaty of friendship with Germany that required mutual consultation in the event of a military attack against either country." [6]

The issue is, Franco also signed a similar treaty with Portugal, which was "harmonized with the traditional British alliance with Portugal", keeping Franco in a more neutral position than at first glance.[7] Franco did align Spain's economy with the Nazis and Italian Fascists, however, which "had disastrous effects on the Spanish economy during World War II" and "made it impossible for Spain to exploit the conomic opportunities of neutrality" as others did.[8] These policies created a great deal of hardship for Spain, and meant they were not in much of a position to fight in 1939 or really any time soon thereafter.

"Though Franco provided some indication of still favoring German policy, the Spanish government had no reasonable option but to declare its neutrality."

"In March I940 the senior generals who made up the army's Superior Council approved a statement by the increasingly critical-minded monarchist Gen. Alfredo Kindelan that declared it impossible for Spain to enter the conflict because it lacked economic support and materiel. They also complained of the internal divisions within the FET and questioned its role as partido unico, declaring that the army was "the only instrument available to orient Spanish politics." [9]

The early success of the Nazi war in France did complicate this attitude, although the capacity for Spain to wage war remained low for some time. The Francoist regime was, at least in rhetoric, an expansionist force, although this was mostly restricted to goals such as regaining Gibraltar from the British (which never transpired) and expanding its conquests in Africa (ideally at the expense of France and Britain, but again this never really happened).[10]

"[After the fall of France] Franco was now firmly convinced of German victory and would remain so, though with diminishing conviction, until mid-1944. On 3 June 1940 he prepared the text of a letter to Hitler congratulating him on the incipient triumph over France and identifying Spain with the German cause, which he chose to define as a continuation of the struggle waged by the Nationalists in the Civil War. At the same time he detailed the economic and military weaknesses that made it difficult for Spain to enter the war at that time, and on the following day Beigbeder handed the German ambassador a list of Spanish claims in northwest Africa."

After this point, the Francoist regime adopted a policy of "nonbelligerence" rather than "neutrality", and although it continued to avoid getting involved in the Italian & German wars, it maintained its support for their actions. The Francoists did discuss possible terms for a Spanish entry into the war with the Nazis, however Hitler dismissed these as he did not think their support was worth the concessions (mostly territorial claims in Africa,[11] and the risk of his armistice with the defeated France falling apart if Spain intervened).[12]

Contrary to post-war Francoist mythology, Franco was greatly tempted to fight alongside the Germans, but Spain was not in a position to do so after the Civil War. Hitler found Franco’s demands for French territories and Gibraltar excessive, especially given the German leader’s aim to bring the French on side. Both dictators intensely disliked each other. Hitler was so annoyed by Franco’s behaviour that he later told Mussolini that he ‘would prefer to have three or four teeth taken out’ rather than have another round of negotiations with the Caudillo, ‘not a man who was up to the job of the political and material construction of his country’. This criticism of Franco was probably also a strategy for Hitler to reassure the Duce that their relationship was special. [13]

More negotiations did take place, but the main roadblocks were, again, the unpreparedness of the Spanish economy and its vulnerability to a British blockade, as well as a general lack of military power.[14]

Hitler wanted Spain to enter the war immediately while merely trusting in German good will to provide a certain amount of economic assistance and military supplies; moreover, he refused to make any territorial commitments on northwest Africa in advance. Ribbentrop was even more aggressive, asking for the cession to Germany of one of the Canary Islands as a naval base, together with one or more ports in any southward expansion of Spanish Morocco. On Franco's orders, this was rejected with barely concealed indignation, and any agreement on entering the war now began to be postponed behind a screen of Spanish economic, military, and territorial requests. [15]

I have presented a view from only a couple of sources here, and they have heavily implied that Franco would have loved to join the war in order to make gains in Africa (and crucially retaking Gibraltar). The main roadblock to that happening was a general weakness of the Spanish economy and war machine.

This is not necessarily the only history of Francoist Spain, however. Whilst territorial gains would have been perfectly in keeping with the goals of the Francoist regime, they were not a priority and joining a suicidal psuedo-holy-war against Bolshevism alongside the increasingly erratic Hitler was never really, in my opinion, on the table. Some historians (I'm sorry I can't find the source now, I thought it was in Goeschel but I couldn't find it) even doubt how genuine Franco's offers to join the war really are, and whether the Caudillo actually knew full well his demands would not be met by Hitler but had to offer anyway to maintain the assistance for his own regime. As with the penultimate quote above from Goeschel, there is a significant amount of "post-war Francoist mythology" about what role exactly the regime played during the war. This can make the historiography difficult, as one must consider the effect of the Francoist regime's own spin on its actions (in order to absolve itself of any deep association with the defeated Axis powers) when investigating its motives.

Hope that answers your question! Source in next comment