Вместо перехода по URL-адресу "/ settings-change" это просто обновление страницы с данными в URL-адресе
$(document).on("submit", "#change-settings", function(e) {
e.preventDefault();
let form = $("#change-settings").serialize();
console.log(form)
$.ajax({
url: '/settings-change',
type: 'POST',
data: form,
success: function(res) {
console.log(res);
window.location.href = "/profile";
}
});
});
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.3.1/jquery.min.js"></script>
<form id="change-settings">
<div class="form-group label-static is-empty">
<label for="profession" class="control-label">Profession</label>
<input autocomplete="off" name="profession" type="text" id="profession" class="form-
control" value="$session['user']['profession']" autofocus="" />
</div>
<button class="btn btn-raised btn-info waves-effect">Update</button>
</form>