Short answer, you're right that it's implied to be the elixir. This was alluded to in "No One Mourns for the Wicked" and the connection is more explicit in the book where Elphaba's mother drinks the Miracle Elixir and immediately thereafter, Elphaba is born with green skin. The characters in the book have their own theories. To quote Wikipedia, "Melena sees the color as a punishment for her infidelity to her husband. Frexspar originally sees it because of his careless words “The devil is coming” on the day of her birth, and later as punishment for his failure to protect his parishioners from the Clock of the Time Dragon." Another popular fan theory is that the green skin is the result of combining human and Ozian genetics. There's also a plant metaphor in how Elphaba doesn't get a chance to grow as a person (and is indeed told that she has no right to grow as a person) because she's kept in the dark.
The Wizard does indeed later offer a draught of it to Glinda to ease her pain. As to why he offers it, it's his patent medicine, a legacy of his history as a snake oil merchant. As is commonly the case for such vendors, he would have advertised it as healing all ills whereas it's sometimes as likely to cause those ills due to the ingredients.