If I can pull it apart a little more, it comes from the Greek "Phobos" which is usually translated as fear, but there's a bit more to it than that.

The Greek god Phobos was the god of fear and panic*

Panic is a bit deeper than just plain old fear, it's a state where your fear is so intense that you're not acting rationally, you're in fight-or-flight mode, you're not thinking through your actions, you're basically running on adrenaline and instincts, and not necessarily good instincts.

A phobia is that sort of irrational fear. It's panic.

I'm a tiny bit afraid of heights (really it's a fear of falling from a high place, but I'm splitting enough hairs here already,) I don't have a phobia of heights. I can approach my fear rationally and overcome it, and even enjoy experiences in high places (I love roller coasters, I'm a hiker and love standing on top of a mountain, etc.) I just have to sort of calm myself down a little and tell myself I'm gonna be ok as long as I pay a little extra attention to what I'm doing. That little bit of fear actually makes it more exciting and enjoyable in its own weird way. If I had a phobia of heights, there's no way in hell I'd be getting onto a roller coaster or going up a tall mountain, even the thought of it might set me off, maybe I'd run away from whoever suggested it, or I'd curl up in a ball hyperventilating, if my friends grabbed me by the arm and tried to drag me onto a trail I might start throwing punches.

To let my nerd flag fly a bit, the "Litany Against Fear" from Dune comes to mind.

I must not fear.
Fear is the mind-killer.
Fear is the little-death that brings total obliteration.
I will face my fear.
I will permit it to pass over me and through me.
And when it has gone past, I will turn the inner eye to see its path.
Where the fear has gone there will be nothing. Only I will remain.

That's about not letting your fear turn into panic. To face the things you don't know and don't understand with a level head instead of going off half-cocked in fight or flight mode.

That's where homophobia fits in. It's not rational, there's no good, logical reason for them to dislike gay people, and they're not approaching it with a clear head. They take this non-issue that they don't understand, and panic about it, they let it consume a ridiculous amount of their (usually very limited to begin with) mental capacity, leaving no room for them to approach it rationally. They can't flee from it, because how do you flee from normal people who are all around you, so that leaves them with fight, because their panic doesn't leave any room in their minds to just accept that gay people don't actually pose any threat to any of the things they're worried about.

Similarly, you get the gay/trans panic "defense" that some people try (unfortunately sometimes successfully) in court to avoid the consequences of their actions, basically claiming that they reason they assaulted or killed a gay person was because they thought they were being hit on by them and they panicked.

With terms like hydrophobic materials, we're kind of anthropomorphizing them a bit. A hydrophobic person is afraid of water. Rabies was known as hydrophobia because people and animals infected with it often had an aversion to water, they'd be thirsty, but one of the symptoms is difficulty swallowing, so when presented with water they'd panic because they wanted to drink but also knew if they tried to they might drown. When it came time to slap a label on materials that won't interact with water, they just repurposed the existing word "hydrophobic" instead of creating a new term.

*the word "panic" itself is actually derived from the god Pan, a god of things like shepherds, nature, music, and having lots of sex. Overall a pretty chill dude as far as Greek gods go, seems like kind of a weird choice to name panic after, so where did that come from? Well Pan liked to take naps in the afternoon. And if something disturbed his nap he'd give a loud, startled shout, which scared all the sheep away. That sort of sudden fear became known as "panikos." You can kind of think of panic as people acting like startled sheep, "sheeple" if you will.

yep. to add: that stuff you can spray on your windshield and the fabric softener you use on your clothes both make water hydrophobic. it doesn't mean that the water is afraid of the substance. the suffix means to be averse to or repelled by something.