Off the bat I should say that I hope someone with more knowledge of Portuguese diplomatic history can delve into the specifics about how Portugal became involved in the founding of NATO, but that much I can say - Portugal was a founding signatory of the North Atlantic Treaty, so it wasn't "allowed" to join NATO as much as it participated in its foundation. Portugal, despite being neutral in World War II, wasn't nearly as isolated internationally as neighboring Spain was, and in some respects its neutrality was stretched really thin (it provided military bases in the Azores to British and American forces during the Battle of the Atlantic). It was already more closely involved in post-war American-led efforts in Western Europe, being a recipient of Marshall Plan aid. Including Portugal in the founding of NATO is not therefore terribly out of line.

But more concretely as to why it being a dictatorship didn't matter - NATO membership criteria including (among other things) a market economy and a democratic government are criteria that were not spelled out until after the Cold War. Until 1999 only four countries joined NATO: Greece and Turkey in February 1952, West Germany in 1955 (the Two Plus Four Treaty of 1990 included former East Germany), and Spain in 1982. These were all essentially on an individual basis and not part of any sort of pre-established membership plan.

Minimum criteria for future members was spelled out in the Madrid Summit of July 1997, which saw invitations extended to Hungary, the Czech Republic and Poland, who then formally joined in early 1999. The April 1999 Washington Summit developed the Membership Action Plan mechanism for future members to apply through, and Eastern European countries interested in joining (the "Vilnius Group") were encouraged to seek membership through this mechanism. Seven of these countries (Bulgaria, Estonia, Latvia, Lithuania, Romania, Slovakia and Slovenia) were extended invitations in 2002 and joined in 2004, with Albania and Croatia joining in 2009, Montenegro in 2017, and North Macedonia in 2020. Currently the only country with a Membership Action Plan is Bosnia.

Sweden and Finland were extended invitations without Membership Action Plans, so even here it shows that this mechanism isn't an ironclad rule. Nor is it an ironclad rule that meeting minimum requirements, submitting an application to join and receiving a formal invite equals automatic membership - as of December 2022 Finland and Sweden's membership status is effectively being blocked by Turkey, as new membership needs to be ratified unanimously by all existing NATO members.