Your this.state refers to an object containing an array,
categories: [...]
So you'd need to map over this.state.categories.map instead of this.state.map
And also, I've noticed that button event handler mutates the state.
<button onClick={() => this.setState(() => (item.isClick = false))}>
  {item.category}
</button>
You'd need to return a new item reference to notify react that the state has been changed as well as adding a key prop.
Thankfully, someone asked a question about the reason for using key (only 2 hours ago) and replied, which you can refer to.
https://stackoverflow.com/a/57838612/4035
<button key={item.category} onClick={() => this.setState({
    find the item by searching for each item in `this.state.cateogies`
  })}>
  {item.category}
</button>;
But then it gets tough to find the right element, so I'd recommend to normalize(meaning change the this.state.categories) too look like following for easier way to set the state.
// From this
this.state = {
  categories: [
    {
      category: "sports",
      ariticles: [],
      isClick: true
    },
    {
      category: "sports",
      ariticles: [],
      isClick: true
    }
  ]
};
// to this
this.state = {
  categories: {
    sports: {
      category: 'sports',
      ariticles: [],
      isClick: true
    },
    news: {
      category: 'news',
      ariticles: [],
      isClick: true
    }
  }
};
This way, you can set the state like following.
<button
  key={item.category}
  onClick={() =>
    this.setState({
      [item.category]: {
        isClick: false
      }
    })
  }
>
  {item.category}
</button>
Reply to the comment below (I've translated the comment for non-Korean speakers)
So you can't use .map any more after making such a change?
I am sorry, I should've updated the category item generation code after the state update.
No, you can't use .map as it's now an object. But thankfully, you can use either Object.keys/value/entries to iterate the this.state.categories.
You can click on "Run code snippet" button to see the output
let state = {
  categories: {
    sports: {
      category: "sports",
      ariticles: [],
      isClick: true
    },
    news: {
      category: "news",
      ariticles: [],
      isClick: true
    }
  }
};
Object.entries(state.categories).map(([id, item]) => console.log(item))
 
 
So your code would look like something like,
<div className="categoryBtn">
  {Object.entries(state.categories).map(([id, item]) => (
    <button
      key={id}
      onClick={() =>
        this.setState({
          [item.category]: {
            isClick: false
          }
        })
      }
    >
      {item.category}
    </button>
  ))}
</div>