Is your silver cord actually your spinal cord?

No, it is the stretching of the soul through another dimension.

The silver cord (from Ecclesiastes 12:6) is a metaphor for the central nervous system. So yes, the spinal cord.

It’s your consciousness.

Short answer: no.

Long answer:
Among the more metaphysically or esoterically oriented, there are many who speak of the silver cord mentioned by the OP. It is often observed during astral travel, or out-of-body experiences, and believed to act like a tether between the “astral body” and the physical body. It is believed to be connected to the physical body through the seven major Chakras, or energy centers in the subtle body.

The spinal cord, on the other hand, refers to the physical bundle of nervous tissue running through the spinal column. It is a long, thin, tubular structure of nervous tissue, and it extends from the medulla oblongata in the brainstem to the lumbar region of the vertebral column.