<div id="app">
  <button class="add-item">Добавить блок</button>
  <div class="menu__list"></div>
</div>

.menu__list {
  display: flex;
  flex-wrap: wrap;
}
.added-block {
  display: flex;
  width: 120px;
  padding: 1em;
  margin: 1em;
  border: 1px solid #CCC;
}
.menu__item {
  margin: 4px 5px;
}

const btn = document.querySelector('button.add-item');
const createBlockHTML = text => `
  <div class="added-block">
    <div class="menu__item ">${text}</div>
    <button>&times;</button>
  </div>
`;
const container = document.querySelector('div.menu__list');
const addBlock = () => {
  container.innerHTML += createBlockHTML('Lorem ' + Math.random().toString(16).substring(2,6));
};
btn.addEventListener('click', addBlock);

container.addEventListener('click', ({ target }) => {
  if (target.nodeName === 'BUTTON') {
    target.parentElement.remove();
  }
});

addBlock();
addBlock();