A great question.
I personally have always seen red shoes as a representation of dream states, as in The Wizard of Oz, and here is a great site which offers up an interesting take on this, including:
...I think they simply serve to show a bond between the characters. It raises a question, what does this mean, why is it important to them? Later, we discover why; she wore them when he saved her from the Kaiju...
...If you think of red shoes, and you care about film, you should already be at Powell & Pressburger. THE RED SHOES is one of their masterpieces, and very well regarded by many other film makers; undoubtedly Del Toro is a fan. [...] If you don’t flash to THE RED SHOES, you should certainly flash to THE WIZARD OF OZ. [...] None of these references are too significant, nods really, but they help establish a sense of style and feel among other things.
...Finally, Hannibal’s shoe… how does that play in? [...] An empty shoe is a powerful reminder of the person who once stood in it, and I think in this way they can touch on apocalyptic imagery without overstating it. [...] The fact is, shoes are surprisingly unique items that say a lot about their owners and empty, they look…wrong. I think, in the end, it is actually as simple as that. Without necessarily tarnishing more – let us say “worthy” – symbols and images, Guillermo manages to harness them to enhance the depth of his own fantasy piece.
For a more 'highbrow' take on the significance of the shoes, might I direct you to this essay, which explores the deeper psychological significance of shoes in art and literature.
...A shoe without its mate, for most of us, is like a low-grade fever — not a major disaster, but definitely a pain. In a movie intent on matching and pairing everything, it’s a sign that something is afoot.
...Of course, footwear as a visual fulcrum is nothing new. [...] In “The Origin of the Work of Art,” Heidegger speaks of these shoes as reflexive “equipment,” instrumentalized forms that mirror the starkly enduring lives of those who wear them...
...As equipment — as instrumentalized forms that re-express the forms of their users — all three are on the same page. [...] Hannibal’s footwear reprises the high-risk but surprisingly low-casualty life he leads as a black-market dealer in Kaiju parts. His opening bravura, his seeming downfall, his post-credits comeback — all of this is externalized and objectified by the fate of his shoes. What is remarkable here is the deep resemblance between the equipment and its user, the fidelity with which one tracks the other and amplifies the other, thanks to the infinite reproducibility of form across an infinity of scale.
...the nesting of self-replicating geometric forms, as posited by Benoit Mandelbrot in 1975, when he coined the word “fractal.” Shoes are a familiar instance of this: a slightly enlarged re-expression of us, copying both the physical anatomy of the feet encased in them and the emotional contours of the person walking on those feet. Hannibal’s shoes are not alone here. Mako Mori’s red shoes, the other star players in the footwear department, also copy her in just the same way. Mako has initially seemed too overwrought, too stuck in her traumatic past, to be trusted with a Jaeger, one of the giant robots that stand between mankind and the Kaijus. In her backstory, as a child wandering alone through the rubbles of Japan (a scene in turn framed by the large shadow of Hiroshima), she comes across as unhinged, wearing one shoe and carrying the other in her hand. But she’s always had both in her possession, just as she’s always had her wits in her possession. She is a together person, as the togetherness of her shoes could have told us all along.
...Resemblance is key. For all its high-tech contraptions and CGI geekery, Pacific Rim remains firmly rooted in a primitive, resemblance-based, analog world, honoring the iterative sameness of the embodied form, rather than the numerical abstraction of the digital universe.