I'm going to toss this out there:
You commented under NudgeNudge's answer:
The intended answer is indeed one word.
This is indeed a word in Russian:
A one-letter word meaning "myself", analogous to "I" in English
which resonates with the opening of each stanza being "I am". It also has some characteristics touched on here:
I am a Russian ay,
ah, in the English way!
A "Russian A" sounds to me like "the letter corresponding to /e/ in Russian"
and "ah in the English way" means "the same letter sounds like /a/ in English"
Now, Я is pronounced /ja/ ("ya"), but that does rhyme with /a/ in English.
Then we have a nice correspondence with:
the booty's ours, hooray!
Because surely the same pirate would say:
That is, Cyrillic Я, looks like Latin R (backwards), sounds like Arrrr... in English
Now we come to the second stanza:
For many not to say,
and all is quite okay!
Where we can apply the semantics of the word, rather than just its phonetics:
Я, "ya" means me, myself. It can't refer to "many" people, only ever one person.
But of course all are welcome to use it.
But here endeth the likeness, because I can't make an argument for opening of the second stanza (the reverse of the phonemic argument above doesn't work), nor for the mythological status of this word¹.
¹ Except, of course, every story has a protagonist....