What does 'relationship with God' mean?

Let’s say two atheists have an unexplainable wonderful feeling in their heart. One decides to become a Christian and credits that feeling as God revealing himself or as a relationship with God, while the other atheist simply says it’s a good, strong unexplained feeling in his heart. Do they have the same feeling? Does calling the feeling two different things affect what the feeling really is? Is it possible that one can have a ‘relationship with God’ but be unaware of it? Where in the Bible does it say that one must call their relationship with God (or Jesus) a ‘relationship with God’ or merely be consciously aware of it?

If I recall correctly, there’s something about Him being jealous in that regard…

There’s nothing in the Bible that states you have to call it a “relationship with God”.

The only statement regarding this, in many ways, is that God is Jealous, and depending on your reading, Jesus calls God Father, and others do as well.
Aside from this, followers are referred to as Children, again, depending on your reading, and to each other Brothers and Sisters.

The “Family” bit is kind of gleamed from a type of reading of the Bible, and it’s fairly popular.
But it’s not demanded anywhere in the Bible.

If you don’t call it a “relationship with God”, must you know what a “relationship with God” is to have one?

That depends on what a person means by relationship.
Like many things in Christianity, and many religions, it’s an existential definition that is used as a term to refer to the instance of the personalization to the adherence and identified connection one has to the spiritual movements of Christianity.

If you were to ask me this question, being that I’m more clinical and analytical than other theists one might commonly find, I would describe the term relationship in scientific terms as one describes the relationship between an atomic nucleus and an electron, or the relationship between a frequency in hz and it’s audible pitch in sound.