To answer your question.

While it is, indeed, true that Greece had a fascist regime after its coup d'etat and the high level of German exports to Greece during the war could have pushed the Greeks to join the Axis, this didn't happen, for a number of reasons.

First of all, the British. Greece and Britain had had a bit of a "special relationship" from Greece's independence up to that point.

Two days after the war was declared, Metaxas told journalists that he is "following the course of action set by Eleftherios Venizelos" (who had taken Greece into WW1 on the side of the Entente, despite objections by the King) and that "Greece needs to coordinate its foreign policy with that of the Power for which the Sea is, like for Greece, not a boundary, but a corridor [...]. "For Greece, Britain is a natural ally and has always been a protector of Greece, sometimes the only protector, and so the win is hers". Of course, this was after the war had been declared, but this was a view held by the Greek government since the days of Venizelos, that Britain was a "protector" of Greece.

Also, the restoration of the king, George II, which had happened before the coup, and who was living in Britain after his exile, was a big pro-British boost. The King, who still had major influence in Greek politics was pro-British, and there was a desire to avoid a split like the "Ethnicos Dichasmos" that had taken place during World War I.

It is important to note that, two years before the coup, during discussions about the Balkan Entente, Metaxas said that "In no case should Greece find itself on an opposing side from Britain, I would consider this a dogma".

Metaxas, fearing Italian ambitions, had already proposed a Greek alliance with Britain in 1938, but the British rejected it, fearing it would displease the Italians. After this, Greece followed a policy of neutrality, until the Italian ultimatum on the 28th of October, 1940, when it had to fight and join the allies or be completely annexed.

Lastly, and I feel I have been writing too much, it is important to consider the territorial ambitions of neighbouring states to Greece.

Italy, under Mussolini, wanted to restore Italian hegemony in the Mediterranean. It had already refused to give back the Dodecanese islands to Greece, and did not really seek an alliance with Greece. Greece was also, then, in an alliance with Yugoslavia, on whom Italy had many territorial ambitions. For the Greeks to join the Axis, Germany would have to either give up on Italy, or force Greece as a "mediator" to give Italy much of what it asked for. Metaxas felt that was unacceptable.

Bulgaria had lost its corridor to the Aegean and the Mediterranean after World War I, and maintained its claims on the territory. It refused to give these up, as well as its claim in Serbia. Though not at the time a German ally, Italy's and Bulgaria's interests aligned, so it's possible, in Metaxas' view, for Hitler to ask the Greeks to give the Bulgarians Thrace back, in a sort of "Vienna Award Mediation".

And lastly, Turkey. After a long and bloody conflict with Turkey during the Balkan Wars, World War I, and the Greco Turkish wars, Greek lands had substantially increased at the expense of Turkey. At the time, Turkey was strongly aligned with Britain and Metaxas felt that, if they were to join the wrong side, the allies would bring Turkey into the war by promising them Greek lands.

So, to recap, if Greece joined the Axis, it may be forced to make some concessions to Italy or/and Bulgaria and the allies would promise Greek lands to the Turks, who would invade. On the other hand, by opposing the Axis, it would get support for its claims by Britain, and possibly gain the Dodecanese or Northern Albania (which was promised to the Greeks after they occupied it but never materialised).

And, finally, there were many totalitarian regimes, especially in Eastern Europe, in 1936. Bulgaria, Greece, the USSR, Turkey, Hungary, Yugoslavia had been under a royalist dictatorship but was in a bit of a limbo then, Romania, Portugal, you get the point. Of these, no one was allied to Nazi Germany in 1936, and only 3 would ally them at all. Every dictator was different and in Greece, though chauvinism was, of course, present, nothing like the German idea of sterilising sick people and "lesser peoples" could be found. Metaxas' reforms were more similar to Franco's later actions, than the ideology Hitler or Mussolini had adopted.

Ideologically, they were close, but land integrity mattered more to Metaxas than a fascist world order.