Isildur may not be the most reliable narrator over here.
Gollum acquired the ring by murdering his relative Déagol and taking it from him. He would later claim that it was a birthday present.
‘The murder of Déagol haunted Gollum, and he had made up a defence, repeating it to his “Precious” over and over again, as he gnawed bones in the dark, until he almost believed it. It was his birthday. Déagol ought to have given the ring to him. It had obviously turned up just so as to be a present. It was his birthday-present, and so on, and on.
The Lord of the Rings - Book I, Chapter 2 "The Shadow of the Past"
Bilbo got the ring by pocketing it when Gollum had accidently dropped it, and then later claimed that Gollum himself had willfully offered it to him as a prize.
To them his account was that Gollum had promised to give him a present, if he won the game; but when Gollum went to fetch it from his island he found the treasure was gone: a magic ring, which had been given to him long ago on his birthday. Bilbo guessed that this was the very ring that he had found, and as he had won the game, it was already his by right. But being in a tight place, he said nothing about it, and made Gollum show him the way out, as a reward instead of a present. This account Bilbo set down in his memoirs, and he seems never to have altered it himself, not even after the Council of Elrond.
The Lord of the Rings - Prologue 4 "Of the Finding of the Ring"
We see that the ring causes whoever acquires it to embellish the story of how they got it, to make it sound like it is theirs by right, not just by happenstance or otherwise.
So it would fit very well for Isildur to not have been the one to actually kill Sauron, but to still later try to convince himself and everyone else that he did. Note also that he words this as a question, dancing around it rather then saying it outright.
This I will have as weregild for my father’s death, and my brother’s. Was it not I that dealt the Enemy his death-blow?
The Silmarillion - "Of the Rings of Power and the Third Age"
So I'd say to take Isildur's statement with a grain of salt.