Vain and vein are sound-alike words with different meanings, or a type of homonym.
Vain can mean ineffectual or having no effect (He tried in vain to get her to notice him) or conceited, showy, or worthlessly proud of oneself (She was so vain she kept looking at herself in the mirror, instead of at him). And in our above context a vein is a blood vessel, carrying oxygen-poor blood back to the heart, which so many of us long to see gently caressing breast skin with their blue tint. Their blood seems cooler to me when running my nose along one (venous blood, not arterial blood coming from the warm heart) but that has nothing to do with homonyms.

Homonyms are fun (break (destroy) brake (stop), whole (complete) hole (cavity in the ground), punch (a hole), punch (a person), breastwork (a defensive fort of moderate height), breastwork (what we all want to be doing right now)) and important to get right (If you break the brake, you cannot stop, etc).