merhaba 2 yöntemle çözebilirsiniz.
1.yöntem
index.html
<!DOCTYPE html>
<html>
<head>
    <meta charset="utf-8">
    <meta name="viewport" content="width=device-width, initial-scale=1">
    <title></title>
  </head>  
  <body>
    <script type="module" src="index.js"></script>
    
    <div id="container"></div>
    <div class="header">
      <div class="menu">
          <div class="menu-item" onclick="changeCategory('tshirt')">Tişört</div>
          <div class="menu-item" onclick="changeCategory('shoes')">Ayakkabı</div>
          <div class="menu-item" onclick="changeCategory('jewelry')">Takı</div>
          <div class="menu-item" onclick="changeCategory('accessories')">Aksesuar</div>
      </div>
  </div>
</body>
</html>
index.js
(changeCategory fonksiyonu global tanımlanabilir)
export function changeCategory(category) {
    console.log(category);
}
window.changeCategory = changeCategory;
2. yöntem
js ile click aksiyonu alınabilir.

<!DOCTYPE html>
<html>
<head>
    <meta charset="utf-8">
    <meta name="viewport" content="width=device-width, initial-scale=1">
    <title></title>
  </head>  
  <body>
    <div id="container"></div>
    <div class="header">
      <div class="menu">
          <div class="menu-item" data-category="tshirt" >Tişört</div>
          <div class="menu-item" data-category="shoes">Ayakkabı</div>
          <div class="menu-item" data-category="jewelry">Takı</div>
          <div class="menu-item" data-category="accessories">Aksesuar</div>
      </div>
    </div>
    <script type="module" src="index.js"></script>
</body>
</html>
export function changeCategory(category) {
    console.log(category);
}

const menuItems = [...document.querySelectorAll('.menu-item')];
menuItems.forEach((item) => item.onclick = () => changeCategory(item.dataset.category) )