It seems to me that inheritance can't be classified as profit or wage. So does it fulfill the definition of rent, or are there some economic activities that don't fit into this classification?
Profit, wage and rent are rewards for inputs used in production, they constitute a decomposition of Income (or the Income-decomposition of output produced)
Inheritance is a transfer, it does not involve production. So it is invalid to try to fit it into the mentioned classification.
Consider now any gift: say I work, earn a wage in the form of consumer goods, and give them to you. The consumer goods I got as a wage is a reward for my labor. Giving them to you is a transfer of already produced output.
In inheritance, this output has already taken the form of assets rather than consumer goods.
Interesting question to think about. Maybe consider inheritance to be income derived in a past time period. For example, rent a father earns and gifts to his son, thus is still rent. It may be notable to note that inheritance as a substantial portion of income disproportionately impacted the landowners. (The resulting conversation is interesting and worth some reading).