Love Is a Matter of One's Own

When I was younger, I once thought that love was a matter between two people, or even three or four people. Later, however, I came to understand that love is a matter of one's own.

- removal of word 'once': I think you are implying that throughout your youth you thought love was a matter between two people, not that when you were younger then was only a single time when you thought that love was a matter between two people

When we have loved someone, we were influencinged and changing each ed by one another.

- You are using two separate tenses before and after the comma that don't go together

Anyway, we can no longer be the persone we once were.

I think 'one' is technically okay here person makes more natural.

We achieve ourselves through love.

achieve ourselves sounds kind of odd. Grammatically okay but it's unclear what you are actually trying to express

We've never loved someone as deeply as we thought we could.

This sentence is grammatically correct but it's unclear what you mean by 'as we thought'. I just added 'we could' on the end but feel free to change it but you need to add something on the end to be clear with what you mean

When we pursue love, we're just searching for the part of ourselves that got lost somewhere.

It sounds a little unnatural to have 'ourselves lost somewhere' without either 'that got', 'that has been' etc in between or you could write '... ourselves, lost somewhere in the ether'

One day, I have to admit that I couldn't love no anyone.

I think your sentence is grammatically correct but it's a bit archaic. More natural in modern English to say 'couldn't love anyone' then could love no one'

Love is not free. Only when we achieve ourselves will it be free.

achieve ourselves is again not clear here

We achieve ourselves through love.

see note below about "achieve ourselves"

One day, I have to admit that I could love no one.

Tense issue. "One day" is ambiguous, so I'm not sure if you're talking about the past ("could") or the future ("have to").

Love is not free.Only when we achieve ourselves will it be free.

"achieve ourselves" isn't a normal combination. Some more common options:
- reach fulfillment
- attain fulfillment
- feel fulfilled
- accept ourselves
- embrace ourselves

Interestingly, the first time you used it, I felt like it sort of made sense in a poetic way. But here, as a conclusion, I think you should be clearer and use a more standard term.