Within the scope of international relations studies/political science, you may find out the qualifications of a state by looking some basic points to define it as democracy or not. The periods of the elections, transparency in electing processes, mechanisms in determination of the candidates, role of the party leaders in decision making etc. would make it easy for your categorization. However, these are all internal factors and unfortunately limited for explaining the whole.
Generally speaking, in a different path, foreign policy of a country also gives clues to the researchers. Shaping of a foreign policy can never be completely partitioned from the internal system and its potential outputs. Even we can claim that the national systems generate the foreign policy behaviours, likewise the reverse. It is a complex relationship between the inside and the outside. Both effect the other one. Both spill over.
Israel, so-called 'unique democracy island' regionwide, carries lots of contradictions in its baggage. Put aside the domestic level. Let me focus on the foreign affairs of Israel in a few cases.
When the Arab spring kicked off, almost all the democracies around the world had supported the unclear but hopeful democratization process in Arab world. These reactions were so normal. In theory, democracies support democracies. They do not fight each other. They negotiate their disputes...However, only one democracy had kept relatively silent during the revolutionary developments in Arab history. Particularly the Egyptian part of the Arab spring witnessed a contradictorial treatment by its 'ultra-democratic' neighbor. With presenting the 'national security' as top priority, Israeli leaders hesitated to pose as democracy promoters. Moreover, Israel secretly supported the Sisi coup against Muslim Brothers in 2013, believed that Sisi would restore the Camp David order in full. Whereas, the democracies are natural democracy promoters. Also, it is just an unwritten rule for states to promote similar regimes around. Israel is irrational in this context.
In the aftermath of Cold War, US government and its 'academic' supporters declared victory over Soviet-style communism on behalf of democracies. Then, 1990's and the following years have been discoursively democracy-promoting term for US government, Israel's proudly 'natural ally'. How about Israel, natural ally of US? How about Israel, unique partner of the Western political values in the region?
Israel plays alone.
Supporting radicalist kingdoms like Saudis or autocratic coup leaders like Egypt's Sisi are the normals of Israeli foreign policy. Further, these normals could be noticed in the near history. Turkey, one of the few democracies in the region, was an ally of Israel once upon a time, only in the period of military tutelage in Turkish political life!
It is the democracy, stupid!