To start, we can start with what is a prophet within religion? In religion, a prophet is someone who communicates with the divine or is capable of understanding and expressing the divine will of God. They are responsible for being the bridge between Humanity and the Divine, however that is interpreted within any religion.
A prophet is chosen by the divine entity and are given exclusivity over the insights provided. No one else is able to communicate with God in the same way, no one else is able to hear what they are hearing, yet it is trusted that it comes from God. In this way it seems that being a prophet is highly intertwined with the existence of God and religion. So how are we going to separate out the role of a prophet from religion and from the divine?
The first part is to try to define the entity with which a prophet communicates from which they are provided universal insight. The question comes down to "What is God?", but that question is one where I cannot provide a concise answer. I cannot speak to what higher powers exist that have not made themselves evident to all of Humanity. What I can do however is define this aspect of a higher power that a prophet communicates with. That is knowledge of the unknown. A prophet is provided insights and truths into what is not known. A prophet is not worth anything if all they tell you are trivialities and things which you already know the answer. A prophet's role is to provide you answers and truths that are unknown to all of Humanity.
With that we can start to see what a prophet is without religion and that is someone who speaks truths that they believe, whether from the divine or not, that are fundamental to Humanity and the world. They work in the world of proclamation and prophecy. They speak to truths without evidence other than from the belief that others have in the source that they provide. If one believes in God as omniscient and as a communicator to Humanity, then if someone says that a message came from God, they are more likely to believe in the veracity of the statement spoken.
This leads us to an important question and aspect of a prophet. How does one become a prophet? If a divine entity speaks only to people that are prophets, then how when someone declares they are a prophet are we supposed to trust that they are not a false prophet? It can be done in one of two ways, but both end up the same, either others are prophets themselves and believe which then propagates to others believing in them as a real prophet or one provides enough truths that others begin to accept them as a speaker of truths and insights, divine or otherwise. Both lead to the fact that a prophet can only become a prophet if they are accepted by Humanity as a prophet. They can believe it themselves, but they have no power as a prophet without others belief.
This is currently how I am a prophet, no one believes I am a prophet in how it is currently understood, in fact I am saying that I speak without any divine backing, my authority only comes from my travels into the abyss. From how I choose to stare into the unknown and dive deeper into what it means to be human. To wrestle with the vastness that is everything that is not known. I strive to know God, I strive to know what cannot be understood, whether God exists or not, or whether they want me to or not. Eventually, others may choose to trust in what I share, but that is for others to do of their own free will.
tl;dr: A prophet speaks of universal truths that they believe are fundamental and unchanging. These truths are ones that cannot be proven in any evidentiary manner, they are truths that require belief. Belief is the only way in which these truths can truly become fundamental and true.