In your Toggle function you are currently only updating the field with the id password. So when you change the type of that field, only that field is changed. You are not checking the other field called retypedpassword, therefor it won’t be changed. Check out the my addition to your code here below for a working version of your toggle for BOTH fields.

function Toggle() {
    const fields = [password, retypedpassword]

    fields.forEach( x => {
        if (x.type === "password") {
            x.type = "text";
        } else {
            x.type = "password";
        }
    });
}

And a second exmample using the style of your current code:

function Toggle() {
    let btt = password
    if (btt.type === "password") {
        btt.type = "text";
    } else {
        btt.type = "password";
    }

    let otherfield = retypedpassword
    if (otherfield.type === "password") {
        otherfield.type = "text";
    } else {
        otherfield.type = "password";
    }
}

Try this. Note I remove the toggle from the password field click and gave the fields a class

I also streamlined your submit handler and fixed spelling and added required

document.getElementById("showpasswords").addEventListener("click",function() {
  [...document.querySelectorAll(".password")].forEach(p =>
    p.type = p.type === "password" ? "text" : "password"
  )
})
document.getElementById("registerdetails").addEventListener("submit", function(e) {
  const errSpan = document.getElementById("passwordmatcherror");
  let firstPassword = this.password.value;
  let retypedPassword = this.retypedpassword.value;
  errSpan.classList.toggle("error", firstPassword !== retypedPassword);

  if (firstPassword !== retypedPassword) {
    errSpan.innerHTML = ("Passwords do not match. Please retype");
    e.preventDefault();
  }
});

<section>
  <div id="formcontainer">
    <form id="registerdetails" action="fma_t3confirm.html">
      <div>
        <label for="password">* Password (Must be 8 characters and include one uppercase, one lowercase and
						one numeric):</label><br/>
        <input type="password" class="password" id="password" pattern="^(?=.*[a-z])(?=.*[A-Z])(?=.*d)[a-zA-Zd]{8,8}$" required>
        <label><input type="checkbox" id="showpasswords"> Show passwords</label>
      </div>
      <div>
        <label for="retypedpassword">* Retype your password:</label><br/>
        <input type="password" class="password" id="retypedpassword">
        <span id="passwordmatcherror"></span>
      </div>
      <input type="submit" />
    </form>
  </div>
</section>

A simple way is just adding another button for the second input:

function Toggle() {
    let btt = password;
    let btt2 = retypedpassword;

    if (btt.type === "password") {
        btt.type = "text";
        btt2.type = "text";
    } else {
        btt.type = "password";
        btt2.type = "password";
    }
}

Add a class pwd in both password fields. Also remove the Toggle method call on click of 2nd password field

<input type="password" class="pwd" id="password" 
                                       pattern="^(?=.*[a-z])(?=.*[A-Z])(?=.*d)[a-zA-Zd]{8,8}$">
<input type="password" class="pwd" id="retypedpassword">

Your Toggle function:
In the Toggle function, get all the pwd fields and then toggle the type of all fileds.

  function Toggle() {
    let pwdFiels = document.querySelectorAll('.pwd');

    for (let fields in pwdFiels) {
        if (pwdFiels[fields].type === "password") {
            pwdFiels[fields].type = "text";
        } else {
            pwdFiels[fields].type = "password";
        }
    }
}

I am trying to find a solution if is possible to show / hide the password at the same time?

I could only show / hide for the first original password.
Does anybody know a solution for it or if is even possible?

Here is JavaScript and HTML code

function Toggle() {
  let btt = password
  if (btt.type === "password") {
    btt.type = "text";
  } else {
    btt.type = "password";
  }
}

function checkPassword(register) { //correct
  let fisrtPassword = register.password.value;
  let retypedPassword = register.retypedpassword.value;
  if (fisrtPassword === retypedPassword) {
    return true;
  } else {
    passwordMatchError.setAttribute("class", "error");
    passwordMatchError.innerHTML = ("Passwords do not match. Please retype");
    return false;
  }
}