Personally, I don't have a good enough intuitive sense of the meaning to decide what the best translation is, which is why I was reluctant to post an answer.
Luckily, I don't have to explain it myself. Let's take a look at an article titled Robotics' Uncanny Valley Gets New Translation:
[T]he first English translation was done between the early morning hours of 1 and 2 a.m. in a Japanese robotics lab in 2005 — a rush job that has finally received a painstaking revision in 2012.
So you may be correct that familiarity isn't the best way to put it. The article goes on to talk about the various translations of the term (familiarity, likableness, comfort level, and affinity), and it goes on to say:
Such English words fail to capture the full essence of Mori's original Japanese, said Karl MacDorman, a robotics researcher at Indiana University who served as one of the English translators for the uncanny valley essay.
"I think it is that feeling of being in the presence of another human being — the moment when you feel in synchrony with someone other than yourself and experience a 'meeting of minds,'" MacDorman said. "Negative 'shinwakan,' the uncanny, is when that sense of synchrony falls apart, the moment you discover that the one you thought was your soul mate was nothing more than smoke and mirrors."
If the article is accurate, then Karl MacDorman doesn't believe any of these English words accurately convey the meaning of shinwakan by themselves. In fact, he's one of the two translators credited on the 2012 revision (The Uncanny Valley, IEEE Spectrum). This new translation has been authorized and reviewed by Masahiro Mori himself, so that lends credence to his words.
So what is the best translation, if not one of the above? Well, it appears that they chose the word affinity, despite its presence in the list! But why, if it failed to capture the meaning of shinwakan?
Ultimately, words mean what you explain them to mean. In this context, affinity means what the translator explained above because they've explained what they meant, and although this may not be exactly what the word usually means, it is nonetheless close enough to be the term that was chosen by the new translators and approved by Masahiro Mori himself.
As a result, I think we can say that affinity is the best translation.