The prefix of "obsess" is "ob-".

People obsess over a billion and one things. Often they obsess about things that are sexual in nature or are things that have tortured the or caused them pain in the past. Some crossdressers obsess about clothing but just because someone crossddreses doesn't mean they obsess about clothes.

She would obsess over every detail of her plan to ensure its success.

"Dave, don't obsess over it," said Luke.

If you really like someone or something, you might obsess over them/it.