There are food items other than flesh of animals. Some people cook such food and give it away while they do not invoke Allah's name on it but invoke some other entity like a Prophet or a saint. Can we eat such food? Or it is ḥarām for us to eat when offered by any?
We eat all edibles after invoking Allah's name. Usually we use bismillah for the purpose. As for invoking Allah's name while slaughtering that is another issue. This does not apply to other edibles like rice, vegetables etc. However, if someone invokes a name other than Allah in such a way that he ascribes the blessing to other than God, he commits polytheism. His sin is graver than merely eating a haram thing. However, it does not render the food haram for others. It can only be judged after carefully analyzing the wording if the statement implies polytheism or not. It also depends on the person making such duas. Sometimes people indulge in polytheistic beliefs but they believe it to perfectly accord to monotheism. Such a person cannot be considered to be a polytheist. Only such a person can be considered a polytheist who openly claims to be the follower of polytheism, a believer in more than one God.