Dice throwing simulation in Java
Welcome to CodeReview. Regarding your issue, a quick solution is:
else if (dice1 == dice2 || dice1 == dice3 || dice2 == dice3) {       
    int doubleNum = dice1 == dice2 ? dice1 : dice1 == dice3 ? dice1 : dice3;
    System.out.println(&quot;Congratulations : You threw TWO - &quot; + doubleNum + &quot;s&quot;);
    System.out.println(&quot;You win $5&quot;); 
}

The ternary operator makes the code more compact, but in this case I think it's a bit less readable than a chain of if-else.
Few other suggestions:

The while-loop can be replaced by a more convenient for-loop. From:
int round = 1;
while(round&lt;=3){
    //...
    round++;
}

To:
for(int round = 1; round &lt;= 3; round++) {
    //...
}


Declaring multiple variables in one line int dice1, dice2, dice3; is not considered good practice in Java.
Instead of generating a random float and then casting it to an int, generate directly an int with ThreadLocalRandom.current().nextInt(6). Docs.
You can print the first line with System.out.format. From:
System.out.println();
System.out.println(&quot;Dice 1 : &quot; + dice1 + &quot;    Dice 2 : &quot; + dice2 + &quot;    Dice 3 : &quot; + dice3);

To:
System.out.format(&quot;%nDice 1 :  %d    Dice 2 : %d    Dice 3 : %n&quot;, dice1, dice2, dice3)


Class names should be PascalCase. Instead of Lab6_part3 you can name it Lab6Part3.
Declare a constant for the number of rounds, to make it easier to change. For example:
public class Lab6Part3 {

    private static final int ROUNDS = 3;

    public static void main(String[] args) {
        for(int round = 1; round &lt;= ROUNDS; round++) {
            //...
        }
}



Another approach is to generate three numbers with Random#ints and calculate the frequencies:
Random r = new Random();
// Generate three random numbers from 1 to 6
IntStream diceRolls = r.ints(3, 1, 7);

// Generate map of frequencies
Map&lt;Integer, Long&gt; freq = diceRolls.boxed()
        .collect(Collectors.groupingBy(Function.identity(), Collectors.counting()));

boolean winner = false;
for (Entry&lt;Integer, Long&gt; entry : freq.entrySet()) {
    int number = entry.getKey();
    long times = entry.getValue();
    if (times == 3) {
        // Three of a kind ...
        winner = true;
    } else if (times == 2) {
        // Two of a kind...
        System.out.println(&quot;Congratulations : You threw TWO - &quot; + number + &quot;s&quot;);
        System.out.println(&quot;You win $5&quot;);
        winner = true;
    }
}
if(!winner) {
    // Did not throw anything of value
}