I need some help here. Is there a difference if I buy the coin with USD, BTC or ETH. If I own the coin does only the value of my coin matter? Or is it connected to the price of the underlying pairing?

For instance I buy ABC coin for $1, or the equivalent at that moment in BTC. And then BTC goes from $18,000 to $15,000, the value of my coin is then worth more BTC if I sell it correct?

I do understand there's some risk in BTC pairings if there's not sufficient volume, so they may be more susceptible to swings because of being thinly traded, but I own the coins right? Not the pairing USD, BTC or ETH?