If a user enters via link to a product, do they have to buy that same product for me to receive the commission?

It is not necessary for the user to buy the exact product from the link you created in order to receive your commission. Any purchase made by the user through any of your links will be related to your affiliate ID and you will receive the commission. For example, if you create a custom link to a sofa and the user clicks on your link to the sofa but ends up buying four chairs, you will receive commission from the sale of all four chairs.