meaning of, "my very self"

I read this:

and said: “I swear by my very self—oracle of the Lord—that because you acted as you did in not withholding from me your son, your only one." [link]

and I don't know how to interpret it. Is it just a stylized way to say "I swear for myself"?

There are two aspects to your question:-

1. The usage of very, about which you have asked, and
2. The use of by about which you have not.

There is a use of 'very', to modify an adjective. It intensifies it. So, if I say "It is very cold in here", it intensifies the word cold. If I call Jane clever and Amanda very clever, you will gather not merely that Amanda cleverer than Jane, but that she is a great deal cleverer.

But very also modifies (and so intensifies) nouns, as you can see from the definition and examples set out in definition c of the Cambridge English dictionary online.

very adjective [before noun] (EXACT) C2 (used to add emphasis to a noun) exact or particular:

• This is the very book I've been looking for all month.
• You're the very person we need for the job.
• What ended up happening was the very thing we feared the most.
• The very idea/thought of having her friends to stay fills me with dread.

In this use it can sometimes be replaced by 'actual', as in these examples. In the passage quoted, the speaker is swearing an oath by him/herself. To emphasise the point, s/he does not say just "I swear by myself" but I swear by my very self, so splitting the my from the self.

Which brings me to the word 'by'. To swear by something is less common that it used to be. But everyone is familiar with courtroom dramas in which the defendant says "I swear by Almighty God that I shall speak the truth, the whole truth and nothing but the truth.". The practice continued today, though enlightened states also allow the defendant to "affirm". the idea is that nobody would dare to swear a false oath to God, because of the eternal penalty that might follow in the afterlife. Similarly, people have been known to swear "by this right hand". Presumably the idea is that they would be willing to have it cut off if they were found to have lied. Swearing "by my very self" is thus a kind of extreme promise. How could anyone possibly promise falsely him/her very self?

I can't speak for the Hebrew original, but the (Greek) LXX has Genesis 22:16 has Κατ̓ ἐμαυτοῦ, which can be given the gloss "by myself". I'm not sure which translation you've used, but the insertion of "very" may be an attempt to overcome the ambiguity of the (English) gloss, an ambiguity which I think isn't present in the original.

Tuffy's answer has addressed the meanings of "very" and "by", so the rest of this answer will focus on the ambiguity of "by myself", looking solely at the English and not the Hebrew or Greek from which your quote may have been translated.

To swear "by myself" can mean different things. Inserting "very" biases the understanding, probably in an attempt to match the original intent.

Let's take a look at two sets of definitions.

self noun 1 A person's essential being that distinguishes them from others, especially considered as the object of introspection or reflexive action. ‘our alienation from our true selves’ -Lexico

myself pronoun
1 first person singular reflexive Used by a speaker to refer to himself or herself as the object of a verb or preposition when he or she is the subject of the clause. ‘I hurt myself by accident’
2 first person singular emphatic I or me personally (used to emphasize the speaker) ‘I myself am unsure how this problem should be handled’
3 literary first person singular Used by a speaker to refer to himself or herself; I. ‘myself presented to him a bronze sword’
-Lexico

The translation could have been left as "by myself" - other translations do that, and it's not wrong to do so. Each word choice has its own strengths and weaknesses. The weakness with "by myself" that is relevant here is that it can communicate any of the 3 senses in the definitions above (well, probably not the third, so we'll leave it at 2 senses):

1. "myself" is the authority that gave power to what was sworn; or
2. the speaker did not rely on someone else to do the swearing

In your quote, "by myself" identifies the empowering authority. The translator could have simply added a space "by my self", which would be understood according to the definition of "self" above. However, if the passage was read aloud, it can be tricky for the ear to distinguish "by myself" from " by my self". Inserting "very" makes it unambiguous.