I did start to reply with a post about changing the elements id names in the properties panel and to make your own custom login and signup lightboxes.
However, then I stopped and replied different as you were talking about a published site and trying to get LastPass to work with it instead.
I can understand the confusion!
Anyways to carry on with Yisrael post above, yes you can only change the element id name in the properties panel from within the Wix Editor.
https://support.wix.com/en/article/corvid-working-with-the-properties-panel
This doesn’t state that you can now also right click on the element to get to the properties panel too if you haven’t got it up on screen already.
Through this properties panel you can also add any event to the element like onClick, onMouse, onViewport and onDblClick which are needed if you haven’t written it through your code.
https://support.wix.com/en/article/corvid-reacting-to-user-actions-using-events
$w("#forgotPassword").onClick( (event) => {
// onClick event is included in the code so no need to click it in properties for element
export function loginButton_click(event) {
// onClick event needs to be clicked on in properties for this element as not included in code.
Also as Yisrael pointed out, if you change your element id name then it needs to be reflected in any code that you write on that page for it too.
Plus, it is always best to change element id names so you can associate it with the function that it is being used for and it is much easier to remember what it is in your code setup too.
Too many time people post on this forum with code that includes a button with a name of button100 or a text box with a name of text200 for example,
Now how can you work with that in your code and trying to remember where and what each 100 buttons do and the same with the 200 text boxes, of which 100+ might not even be used anymore and had been deleted!
Finally, if you need any help with the custom login and signup lightboxes then just post again or ask another question in the forum again.
keep the Wix API reference section bookmarked too as you will need it many a times.
https://www.wix.com/corvid/reference/APIsOverview.html