Every storage device has a capacity. That capacity is defined by how many spaces that the storage device has avalible to store a 1 or a 0 to (aka "bits"), right?

So, for examplle, let's say we have a storage device with 4096 bits. On an 8-bit system, this would mean it has 512 bytes, right? Therefore, I could store a 512-byte text file to it.
On a 32-bit system, that would mean there are 32 bits in a byte. Therefore, the same storage device would only have 128 bytes, right?
On a 64-bit system, there are 64 bits in a byte. The same storage device would have 64 bytes of storage.

So why does my 8 GB flashdrive show up as 8GB on both 32-bit and 64-bit systems? Is there a standard for how many bits are in a byte for storage devices?

The number of bits in a byte does not change depending upon the OS -- there are always 8 bits in a byte.
What does change from 16 bit OS to 64 bit OS is word size (well, register size dictates this).

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I agree. Bits are still bits. Words are different length but the amount of bits remain constant.

No storage is sold as so many words in length.

A byte is always 8 bits, that never changes, it's the definition of a byte.

The OP is using wrong terms. A word length of 8,16,32 or 64 is not termed bytes. It is word length. I used to work on a 40 bit word length system.