Chains are present throughout the whole game as old remnants and possibly as part of machinery
However, there's places where they show to have a deeper meaning
Shaded "The True Anoited" Citadel, Holy Ground where memories are cherished, has an unique style of column that incorporates a chain motif
it can be better appreciated on the ruins immediately around the shoreline gate, where they're better iluminated
This pattern is also present on the cabinets
This motif runs older though, much much older
to the time of the Depths
A statue of a humanoid, in the classic stance of an echo, chest binded in chains
A head wrapped in chains, to the point of covering all features
...but with an X breaking the pattern, cutting through the chains
In a way, the progression of the higher levels of karma look more like cutting through a knot or a ring, a visual of breaking the cycle
this already gives in the meaning behind this motif: the chains are the cycle, binding the self to the carnal, ascension being the severing of the cycle to move freely above and beyond
Its entire memory is filled with a mantra repeated... 5061 times - and then a termination verse. It was worn as an amulet, probably together with many identical others forming a pattern on some garment. The repeating mantra is important because it symbolizes the cyclical nature of life and death, and the termination verse is a symbol for ascension above and beyond it. I don't know how familiar you are with the nature of life and death, but I imagine like all living creatures you have some intuitive knowledge? Then you know that death isn't the end - birth and death are connected to each other like a ring, or some say a spiral. Some say a spiral that in turn forms a ring. Some ramble in agonizing longevity. But the basis is agreed upon: like sleep like death, you wake up again - whether you want to or not. This is true for all living things, but some actually break the cycle. That doesn't apply to you or me though, you are too entangled in your animal struggles, and for me not breaking that cycle is an integral part of the design. Our mantras keep repeating.
"(...) and then a termination verse"
What is a chain if not a number of interlocked rings? The Shaded Citadel pattern even looks more like a double spiral, a double helix
The urges are also described as "binding"
fun fact one of the elements in an echo's design is called chains
it's the row of golden flakes bound to the legs
considering the depths statues, maybe part of the rituals in preparation for ascension included binding the body in chains, as a metaphor for the binding nature of the cycle, meant to dissolve in the void alongside the body as signifier of the liberation of the mind