As an non-Arabic speaker interested in the Qu'ran I'm interested in how this sounds to a native speaker: what is the phrase actually saying?
Is it saying that all praise belongs to God, that the sayer is giving the praise to God (which is how the English sounds), or that the praise is already God's?
That's an interesting question that might best be analyzed by a scholar, but I can give some commentary. It can be both but the primary meaning is that it is Allah who is worthy of all ultimate thanks and praise. Even if we thank and appreciate our employer or friends or family, we know that they are creations of Allah and that no harm or benefit comes to us except by the Power of Allah. But, as well, all of creation belongs to Allah, so it is not untrue to say the praise itself actually belongs to Allah, though I do not presume this to be the primary point of saying Alhamdulillaah. And Allah knows best. This includes more nuance: https://islamqa.info/en/answers/146025/the-difference-between-hamd-praise-and-shukr-thanks