TLDR:
How do you go about denying the existence of god(s) in religions other than your own? Same thing for atheists.
And why are you assuming it implies anything other than what they literally said?
The rest of the answer:
There are thousands of religions and most of them have at least one god. If you are in a monotheistic religion, your disbelief in the number of gods is only different than mine by 1. Do you have an understanding of every god you don't believe exists?
God isn't a proper noun, to me, as if it's a name. It's a word to describe an idea. Me saying "God doesn't exist" is the same thing as me saying "Your god doesn't exist" or "Your gods don't exist". The word "god" is capitalized simply because it's the beginning of a sentence. I could say "GOATs don't exist", while verbally you would hear "Goats don't exist", and you'd probably think I was referring to the barnyard animal, instead of the acronym*.
Also, if we are in a conversation where you are trying to tell me about your religion and I say "God doesn't exist", I'm using the assumption that you know I'm talking about your god. If we were having a conversation about Jeff Goldblum, and I said "Jeff isn't really that good an actor", you would understand that I was referring to Jeff Goldblum, not Jeff from work or Jeff your neighbor. I wouldn't even need to have an understanding of your god to deny it's existence, let alone a belief in it, as you wouldn't have had time to explain it to me.
To me and many atheists, the whole idea of a god makes absolutely no rational sense in reality. When someone talks about a god to us, it's like talking about myth and magic. Trying to describe your god to me doesn't change my idea that they don't exist. There are plenty of science fiction and fantasy stories that go into great depth about their universe and people, but that doesn't make them real. Does someone explaining the Spaghetti Space Monster to you make it real? Do you have to believe in it to say it doesn't exist?
In fact, I consider the Bible and other religious texts to be historical fictions. For instance, the Bible makes wide use of stories from many other religions to fill it's pages. Sure, there are parts of it that can be corroborated in secular history, but so can the movies "WindTalkers", "Apollo 13", "Bohemian Rhapsody", and "Gladiator", but no one thinks they are 100% factual in every movement and word. Even as good as "Apollo 13" is, the Mr. Coffee machine referred to when talking about how much power the LEM had to get the astronauts back to Earth hadn't been invented yet. And "Bohemian Rhapsody" used the original film of the "Live Aid" concert to get as accurate as they could, but a side-by-side comparison shows they aren't 100% accurate. But even as accurate as these movies may be in parts, they are still fiction.
For example, the biblical story of Noah stems from an ancient Sumerian story of a family of river merchants caught in a yearly flood. Otherwise, you'd have to believe that the surviving Noah family went through a period of significant incest to repopulate all of humanity.
Also, I personally believe that Jesus went to India, maybe even Tibet or China, and learned Buddhism, then brought it back and rephrased it in terms his Jewish friends and countrymen could understand. The story of Jesus in the desert where Satan tempts him sounds a lot like the story of Buddha and the Bodhi tree. There's other references that sound like Tao, the Eight Fold Path. So when Jesus refers to god, he may be talking about Buddha, who isn't a god and did actually exist. But I digress.
Bringing it back to the original topic, the Hebrew/Abrahamic god is referred to as Allah, in Islam and some sects of Christianity, as a proper name, so it needs to be capitalized. The Bible says that god has proper names, but they are rarely used. God is more of a title, like Lord, than a proper name. It's simply regular usage that has people believing and/or using god as a proper name, like The Bard.
Thor doesn't exist. Ra doesn't exist. Santa doesn't exist. Quetzalcoatl doesn't exist. Do you have to believe in any of them to agree with me that they don't exist?**
Ok, so Thor does exist. He's all over the Marvel Universe movies and comic books, right? And his real name is Chris Hemsworth.
* BTW, how do you know that someone is "The GOAT"? Maybe there was someone in past history that was far superior and you just don't know it. And saying "of all time" implies that no one will ever be better than them in the forever future. Taken literally, that's an extreme statement and ignores all the performance enhancing medicines and genetic diversity of the future. And if we aren't alone in the universe, some other species may make that person look like an amateur.
** Yes, I said Santa, not Satan, and on purpose. I was going to include Satan, but wondered how many people I could catch if I didn't.