That's what the word "copy" means. If you flip a coin and copy that bit, you will observe that those copies do not behave independently either. If you want independent bits, flip two coins.

Similarly, "erasing" a (qu)bit technically consists of performing a exchange operation between it and a known-zero bit. In typical electronic computers, this would generally involve diffusion-like exchanges between the voltage level in a memory capacitor (such as a FET gate) and that on the GND rail, which has a much greater effective number of bits and therefore will stay mostly zero, but eventually requires a thermodynamic expenditure of known-valued bits (aka negentropy) from some external source to maintain its voltage level / bit zeroness. (This is rather simplified; there's lots of other sources of known-zero and known-one bits getting depleted and replenished, and the exact accounting depends on how you interpret various physical states information-theoretically.)