Javascript(Node.js) solution :

function stringSimilarity(s) {
    let n = s.length;
    if (n === 0) return 0;

    // The similarity of the string with itself is its own length.
    // We start the sum with this value (s compared with s).
    let sum = n;

    // The Z-array will store the length of the longest common prefix 
    // between s and the suffix starting at i.
    let z = new Array(n).fill(0);
    
    // L and R define the interval [L, R] which is the Z-box 
    // (the segment matching the prefix of s).
    let L = 0, R = 0;

    for (let i = 1; i < n; i++) {
        // If i is outside the current Z-box, do a naive comparison
        if (i > R) {
            L = R = i;
            while (R < n && s[R] === s[R - L]) {
                R++;
            }
            z[i] = R - L;
            R--;
        } else {
            // If i is inside the current Z-box, we can use previously calculated values
            let k = i - L;
            
            // Check if the value at k fits within the remaining box
            if (z[k] < R - i + 1) {
                z[i] = z[k];
            } else {
                // Otherwise, we need to extend the search beyond R
                L = i;
                while (R < n && s[R] === s[R - L]) {
                    R++;
                }
                z[i] = R - L;
                R--;
            }
        }
        sum += z[i];
    }

return sum;

}

import java.util.Scanner;

public class StringSimilarity { public static void main(String[] args) { Scanner scanner = new Scanner(System.in); int t = scanner.nextInt(); // Number of test cases scanner.nextLine(); // Consume newline

    while (t-- > 0) {
        String s = scanner.nextLine(); // Input string
        int[] z = zFunction(s);
        long sum = 0;

        for (int i = 0; i < s.length(); i++) {
            sum += z[i];
        }

        System.out.println(sum); // Output sum of similarities
    }

    scanner.close();
}

// Function to compute Z array for a given string
private static int[] zFunction(String s) {
    int n = s.length();
    int[] z = new int[n];
    z[0] = n;
    int l = 0, r = 0;

    for (int i = 1; i < n; i++) {
        if (i <= r) {
            z[i] = Math.min(r - i + 1, z[i - l]);
        }

        while (i + z[i] < n && s.charAt(z[i]) == s.charAt(i + z[i])) {
            z[i]++;
        }

        if (i + z[i] - 1 > r) {
            l = i;
            r = i + z[i] - 1;
        }
    }

    return z;
}

}

**This is the best i could do , it gives the accurate answer but it failed only the time related test case, if some one could optimize this code plz go for it **

def stringSimilarity(s):
# Write your code here
c = 0
for i in range(1,len(s)):
a = s[i:]
for j in range(len(a)):
if a[j] == s[j]:
c += 1 
else:
break
return c + len(s)

!/bin/python3

import math import os import random import re import sys

def stringSimilarity(s): n = len(s) z = [0] * n z[0] = n l, r = 0, 0

for i in range(1, n):
    if i <= r:
        z[i] = min(r - i + 1, z[i - l])
    while i + z[i] < n and s[z[i]] == s[i + z[i]]:
        z[i] += 1
    if i + z[i] - 1 > r:
        l, r = i, i + z[i] - 1

return sum(z)

if name == 'main': fptr = open(os.environ['OUTPUT_PATH'], 'w')

t = int(input().strip())

for t_itr in range(t):
    s = input()

    result = stringSimilarity(s)

    fptr.write(str(result) + '\n')

fptr.close()

import java.util.Scanner;

public class StringSimilarity { public static void main(String[] args) { Scanner scanner = new Scanner(System.in); int t = scanner.nextInt(); // Number of test cases

    while (t-- > 0) {
        String s = scanner.next(); // Input string
        int n = s.length();
        int[] z = zFunction(s);
        long sum = 0;

        for (int i = 0; i < n; i++) {
            sum += z[i];
        }

        System.out.println(sum); // Output sum of similarities
    }

    scanner.close();
}

// Function to compute Z array for a given string
private static int[] zFunction(String s) {
    int n = s.length();
    int[] z = new int[n];
    z[0] = n;
    int l = 0, r = 0;

    for (int i = 1; i < n; i++) {
        if (i <= r) {
            z[i] = Math.min(r - i + 1, z[i - l]);
        }

        while (i + z[i] < n && s.charAt(z[i]) == s.charAt(i + z[i])) {
            z[i]++;
        }

        if (i + z[i] - 1 > r) {
            l = i;
            r = i + z[i] - 1;
        }
    }

Read the number of test cases and input strings. For each string, compute its Z array using the Z algorithm. Sum up the values in the Z array to get the similarity. Output the similarity for each string. Repeat for each test case.

    return z;
}

}