The answer to the riddle "something you can keep after giving it to someone else" most commonly refers to "your word." The English idiom "keeping your word" means to keep a promise made verbally to someone else. The phrase also connotes someone can "take you at your word," or when someone says something, that is precisely what the person promises to do.
For example, someone agrees to pick up someone from work every day at 3:15 p.m. If that person arrives at 3:15 as promised, that person's word has been kept. Contrarily, if someone "breaks their word," a promise broken. If the person fails to pick up the worker at 3:15 p.m. because of other circumstances, that person's promise is broken.
"Keeping your word" can have legal connotations with regards to a contract. In this case, keeping someone's word can have legal ramifications if a written agreement is not kept.
Another answer to the riddle is "a cold" or "communicable diseases." When someone has a cold or influenza, that sickness can be passed onto another person. When both people are sick simultaneously, the first person who has the cold keeps it and subsequently gives it someone else. Communicable diseases spread from one person to another, either through direct person-to-person contact or through food and water. The common cold, influenza, hepatitis, and HIV are all examples of diseases that are passed among humans.