I first encoutered this story well into adulthood, and could never quite articulate why it struck such a chord in me (and clearly, many others).

Later in life, after becoming familiar with the work of Joseph Campbell (Hero with a Thousand Faces) and Jordan Peterson (c.f. his lectures that unpack Pinnochio, The Lion King and his Biblical series), I began to understand the underlying elements in the story that make it reach so deep into our psyches.

Theme 1: pure love as a purifying crucible. Love as the ultimate goal and the ultimate sacrifice. The boy loves the rabbit, but it goes from being new and shiny to being torn and ragged. But love makes it "real". In that sense it is not a sacrifice, but the achievement of a higher ideal.

Theme 2: rebirth and redemption. The Phoenix story. The rabbit is burnt, shedding everything but its purest essence. The tear symbolizes the remnants of attachments to its past self.