It would appear to be Fantastic Elves and where to find them by evansentranced, on fanfiction.net indeed.
After the Dursleys abandon six year old Harry in a park in Kent, Harry comes to the realization that he is an elf. Not a house elf, though. A forest elf. Never mind wizards vs muggles; Harry has his own thing going on. Character study, pre-Hogwarts, NOT a creature!fic, slightly cracky.
Harry runs away or gets left outside after finding a book in a library on magic
From chapter 2, "Wherein Harry takes leave of Kent":
"I..." Harry looked up at her with big, hopeful eyes. "I want to read a story about people who can disappear."
Harry had heard her talking about fairy princesses a couple weeks ago with a little girl, so he knew she wasn't too likely to get angry at him for mentioning unnatural things. All the same, he held his breath after asking the question and watched her purse her lips with some trepidation.
"Like magic, you mean," she decided, nodding. Harry nodded along, and was relieved when she smiled at him. "We have some stories like that. Do you like comic books?"
Meeting odd people along the way, [...], I think Mr. Lovegood, child Malfoy
Xenophilius Lovegood appears in Chapter 7, "Wherein Harry delivers Dirigible Pums".
The man turned in another slow circle and dropped to his knees. "You're safe," he said. Harry narrowed his eyes and disappeared to a lower branch of the tree, silent and suspicious, but still very curious.
"Can you speak English?" the man asked in a soft voice. "My name is Xenophilius. I'm looking for Dirigible Plums. I heard the Forest of Dean has a lot of them if you know where to look. I only want to examine them, and perhaps take a few seedlings, if that's alright."
Draco is without much surprise the Pureblood in chapter 8, "Wherein a pureblood is sighted" (note that once again Harry is convinced to be an elf):
The boy frowned at him, a bit of white hair peeking out from beneath his cap. "What's your name, then?"
Harry cocked his head again. "Elves don't just tell anyone their name." He had found this to be effective in deterring determined name-askers in the past. Harry didn't really see a use for names. He thought it was a very stale thing, to want to label every creature and item that crossed his path.
"My name's Draco," the boy told him, looking a bit perplexed. "Draco Malfoy."
Found with the Google query
harry potter "i am an elf" site:fanfiction.net. The first result is The handbook of Elf psychology which is the sequel of this one. In the sequel, Harry goes to Hogwarts and is still much convinced he's an elf.