mjhjmtu

I am a Russian ay,

ah, in the English way!

For many is to say,

the booty's ours, hooray!

I am an English ay,

ah, in the Russian way!

For many not to say,

and all is quite okay!

What mythological name do we form?

Hint:

My answer is:

The Golden Axe fable, also known as The Honest Woodcutter?

I am a Russian ay,

ah, in the English way!

For many is to say,

the booty's ours, hooray!

In English, "Ay" can be a synonym for "Ah", as an expression of grief (or wonder).

If we interpret "ay" as written in the Russian alphabet, would be written as "au" in English, which is of course the chemical symbol for gold, which is something you can find in a booty.

I am an English ay,

ah, in the Russian way!

For many not to say,

and all is quite okay!

In Russian, the meaning of "ah" can be expressed using the word "ax" (pronounced "akh" in English).

If we interpret "ax" as written in the English alphabet, it means the woodcutting tool, of course.

So since "The Golden Axe" is an old Greek fable, that would fit the mythological answer (Hermes is in the story, apparently). Not sure if it's correct though, because of the "word" tag.

I'm going to toss this out there:

Я

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!

In particular,

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:

Я, matey!
_{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....}

Odin. According to Norse mythology Odin had one eye, hence the pirate reference in "booty is ours, hooray". "Odin" in Russian means "one" which can't refer to "many", hence "for many not to say".

Ay) Ау)
https://ru.wikipedia.org/wiki/Дядюшка_Ау


Возможно с мифологией связан главный герой мультика Лесовичок


Translation:

Uncle Ay (Au). A Russian doll movie trilogy.
Possibly connected to mythology through the main character of the move Lesovichok (Forrest man)