If bitcoin is a share then so is fiat. They are both currencies, only one is tangible and one is digital. In some cases bitcoin can be tangible, but in general fact stocks are simply owning part of the company.
You own a share you own a part of a company, you own a bitcoin you own a piece of currency like you do with cash.
This is a much more accurate description of what bitcoin is then how the OP describes bitcoin.
Owning bitcoin is like owing dollars, or euros, or yen. It is a currency.
+1
I would also add that unlike a stock bitcoin does not need your money. If I bought a stock then the price dropped 50%, I might sell at a loss. That is because below a certain valuation the company may become insolvent and collapse. That is not the case with bitcoin which does not need your money and can continue to operate no matter the price.
With a stock you are sharing profits. With BTC you own something of value that can be spent across the world.