Hi, welcome to the forum, Latex (white sap from tree) is a natural polymer of isoprene but it also contains smaller percentage of proteins, fatty acids, resins and inorganic materials (salts). Polyisoprene when prepared in lab is known as synthetic natural rubber. Rubber and latex is used interchangeably but they are not same. Latex contains around 40-50% of natural rubber. Hence, rubber can be isolated in its pure form from latex.
Latex (not much with rubber) is known to be associated with allergic contact dermatitis, secondary to exposure to allergen and mediated by IgE antibodies. One's body is known to hypersensitive to certain things. Esp. the lips when it comes in contact with the allergen. In such cases, skin allergy test can be useful. Take acre and regards.
Not all rubber is latex. An eraser is rubber but not latex. In the US if a product is made with latex it has to say latex specifically. I personally would not worry about it.
Take care.