It's an informal humorous expression for a romantic relationship. It's quite common in the informal Russian to use rhyming expressions, especially when we want to degrade some concept. For example, "Пьянка-гулянка" is a party with lots of drinking. Some expressions are longer than that. A common way to anwer to the question "Где твой муж? " is "Объелся груш " ( has overeaten pears). So yes, морковь means a carrot, but we don't necessarily imply any Freudian connotations when we say it - it's just a rhyme:)

as they say in russian is a saying that just love at first sight

novel twist have a relationship with someone

Любовь-морковь - billing and cooing

It's a carrot lover :P