Is connotation subjective or objective?

Subjective.

Connotation is how a word makes someone feel or the image it conjures up. Therefore, it varies slightly based on who's talking about it, making it subjective. Denotation, on the other hand, is the dictionary definition (you can remember because both denotation and dictionary start with D), which makes it objective.

For example, childish, childlike, and youthful all mean basically the same thing. However, describing someone as childish makes them sound immature (negative connotation), describing them as childlike makes them sound innocent or pure (positive connotation), and describing them as youthful makes them sound lively and energetic (positive connotation).