Clear the hash before assign new hash. window.location.hash = '';

if (search.toLowerCase() == "about") {
    window.location.hash = '';
    window.location.hash = '#about';
    document.getElementById("search-box").value = "";
    }
else if(search.toLowerCase() == "skills") {
    window.location.hash = '';
    window.location.hash = '#skill';
    document.getElementById("search-box").value = "";
    }
else if(search==""){
    alert("Search Field is empty. Type anything");
}
else{
    alert("Not Found");
    document.getElementById("search-box").value = "";
    }

The problem is, that the hash (#about) don't change when you search for the same hash twice. The browser will only scroll if the hash changes.

I think the best way is search for the element when you klick on the search button and get the offset of the element and then scroll to that position:

Get the Offset of the element:

element.getBoundingClientRect().top + document.documentElement.scrollTop

Scroll to Position

window.scroll(x, y)

As @Balaji Sivasakthi already answered, I would like to point out that you repeat a lot of code.

1. If you want to clear the search box, you don't need to write that in every single if statement. You only need to write it once outside the if statement.
2. If you save search as a variable, you can use that variable for getElementById.
3. Save that element as a variable as well (element below), and you can check if it exists.
4. Declare a message variable, and put an alert outside of the if statements. If message has any value apart from "", trigger the alert.

function Focus() {
  var searchBox = document.getElementById("search-box")
  var search = searchBox.value.toLowerCase();  // 2
  var element = document.getElementById(search); // 2 & 3
  var message = "";  // 4

  window.location.hash = '#';

  if (element) {  // 3 -- checks if you found any elements with id #{search}
    window.location.hash += search;  // adds element id to hash
    console.log({search});
  } else if (search) {  // 2 & 4 -- anything else than "" is true
    message = "Not Found";
  } else {
    message = "Search Field is empty. Type anything"
  }
  
  searchBox.value = ""; // 1
  
  if (message) { alert(message) } // 4 -- anything else than "" is true
}

<input type="search" id="search-box" placeholder="Search">
<button type="submit" id="searchbar" onclick="Focus()">Search</button>



<h1 id="about">About</h1>

<h1 id="skills">Skills</h1>

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