To "like" something has different connotations to "up-voting" something. They are similar and in many ways overlapping but you should choose the one that suits the mental model you want your users to have of your system.
- Likes tend towards being a property of a user. i.e. a user is
recording their personal preferences.
- Up-votes tend towards being a property of a content object. i.e a user is adding a vote to the
You could imagine a system that has both. Answering why - in a system with both - a customer would choose to one or the other might help you choose.
As your system is anonymous I'd probably lean towards choosing "Up-vote"