It took several months of googling ;) but I believe you are
>! Albert Einstein

*Someone broke my heart,*
>! OP mentioned in a comment that this means that the answer has been broken down into parts.

*Part of me resides in the intersection of Rand al'Thor and the 26*
>! Rand al'Thor himself recognized that the intersection of his name and the name of the alphabet is AL

*Sometimes I just need every way of knowing who the fourth to win a race was.*
>! The fourth element on the Periodic Table is Beryllium whose symbol is Be.

*Then again sometimes I just run marathons and win silvers and bronzes.*
>! This could clue RT for Rotten Tomatoes, a website that reviews movies that, as the name implies, do not win gold medals. This has nothing to do with marathons (other than maybe a movie watching marathon) but OP said in the comments that a lot of the words are "gibberish" to "maintain the flow".  The next line seems to confirm that this is in fact about movies.

*But the sums of my life is the sums of all the races I ran and all the movies I saw.*
>! This confirms that the previous clue was about movies and confirms that the answer is
>! AL + BE + RT = ALBERT

*And that gives me the energy to go ahead and fulfil my dreams.*
>! Albert Einstein discovered the famous equation $E = mc^{2}$ which relates energy to mass and the speed of light.

*Who am I?
I've been asking the same to myself for over a hundred years lmao*
>! Albert Einstein was born in 1879 and has had over a century to question his identity. If this answer is correct then I hope it helps mend his heart and reaffirm his identity.

The hints:
>! Albert Einstein had *White hair* and did [tag:physics].