Why was Ethiopia supported by the Nazis and Japanese in the Second Italo-Ethiopian War?

Supported? I think you have it all wrong.

Italy, Germany and Japan formed the Axis power. Why would they support an African country (given the Nazi “Arian” philosophy and all”) AGAINST their own ally?

It is Britain who supported Ethiopia, and that too, only later in the war. In the beginning, when the Ethiopian Emperor Haile-Selassie I made an appeal to the League of Nations regarding Italian invasion, they made a joke of him (save only Russia), but his speech proved prophetic and Britain enrolled its support after the break of WWII. Germany and Japan have not provided direct support to Italy in its war in Ethiopia, or any of its campaigns in East Africa for that matter. Nonetheless, they were allies all along.

At the time of the war, Italy was still neutral in the struggle between the West and Germany. When Hitler tried to takeover Austria in 1934, Mussolini mobilized troops and demanded Hitler stand down. And Italy was still torn between standing with Germany, which she viewed would allow her more conquests, or standing with France, which she felt was much stronger.

Therefore, it was hoped by Germany that the war would aid in weakening Italy, so Austria would be ripe for the taking. The Ethiopian army was pretty poorly equipped, so it was hoped that by supplying rifles to them they could put up more of a fight.

Quite the contrary. Mussolini, who was Hitler's ally, invaded Ethiopia.

Hitler did not support Ethiopia at all

Hitler did not want Mussolini invaded Ethiopia until Hitler did not want Mussolini invading Ethiopia until he had a chance to raid its relics for himself including the Ark of the Covenant before he went to Antarctica