I'd like to have a test that would predict whether two people will get along (enjoy conversation and possibly become friends) if they try? The idea is simple: many people pass the test and then, by looking at results, we can find matches for potential friendship. However, I feel that it's not easy to implement if at all possible.
It's obvious how to find shared interests, but I doubt it would be enough to find a match. So, the test should explore some personality characteristics. The question is what should we test for and how to find matches based on results?
I don't have education in psychology or something similar, and I couldn't find any relevant information. There are many articles regarding compatibility for romantic relationship, but that's not the same thing.
I spent a short period studying and developing conflict and cooperation models, the resulting research was proprietary so I was hesitant to actually respond to this question as there is nothing I can think of to reference directly except maybe some of the implications of the robbers cave experiment (the utility of this experiment is disputed). That being said, I don't believe I ever found a "test" that would predict the likelihood that any two participants would be friends. However, our previously mentioned research found that one of the most important indicators, not a test, but indicators, of potential friendship was not common interest, but common enemies or disagreements. though this answer may not be accepted as it does not meet standard requirements for acceptance, I felt that it was reasonable enough to mention if it helped point your search in the right direction.
There is no such thing as a test for compatibility in a friendship. There are measures for amity (that is, how much we feel someone is our friend), but no tests that predict whether two people would become good friends.
There are tests that aim to test for compatibility in romantic relationships. These are based on theories about what makes romantic relationships work, and some of them are employed on certain online dating platforms. Some of these platforms (e.g. Parship.de and Edarling.de), conduct studies to test their "scientific dating" principles and publish them. What is interesting in this context is that while there are clear indications for what makes a majority of romantic relationships work, but psychologists have as yet been unable to predict what makes people fall in love. So we know what would make them happy in the long term, but we don't know how to get them together. People fall in love for reasons that are very individual and cannot be systematized, yet.
In friendships, there are of course similar predictors as in romantic relationships, but since friendships are often much looser, friends spend very much less time with each other, and therefore don't need the same level of compatibility (because much incompatibility is compensated by less expectations and more time without each other), it is often enough for friends to share the same hobby, the same job, or live in the same neighborhood.
If you think of friendship as very intimate relationships with much sharing of emotions and thoughts, that is, of close "best" friends, then tests for romantic relationships would very likely cover aspects of these friendships as well.