Can multiple different HTML elements have the same ID if they're different elements?

Question

No.

Element IDs should be unique within the entire document.

I think there is a difference between whether something SHOULD be unique or MUST be unique (i.e. enforced by web browsers).

Should IDs be unique? YES.

Must IDs be unique? NO, at least IE and FireFox allow multiple elements to have the same ID.

Can multiple elements have the same ID?

Yes - whether they are the same tag or not, browsers will render the page even if multiple elements have the same ID.

Is it Valid HTML?

No. This is still true as of the HTML 5.1 spec. However, the spec also says getElementById must return the first element with the given ID, making the behavior not undefined in the case of an invalid document.

What are the consequences of this type of invalid HTML?

Most (if not all) browsers select the first element with a given ID, when calling getElementById. Some libraries that find elements by ID inherit this behavior, while newer libraries (as gman points out in his answer) will use the more explicit querySelector and querySelectorAll methods, which unambiguously select either the first or all matching elements, respectively. Most (if not all) browsers also apply styles assigned by id-selectors (e.g. #myid) to all elements with the specified ID. If this is what you expect and intend, then there are no unintended consequences. If you expect/intend something else (e.g. for all elements with that ID to be returned by getElementById, or for the style to apply to only one element) then your expectations will not be met and any feature relying on those expectations will fail.

Some javascript libraries do have expectations that are not met when multiple elements have the same ID (see wootscootinboogie's comment about d3.js)

Conclusion

It's best to stick to the standards, but if you know your code works as expected in your current environments, and these IDs are used in a predictable/maintainable way, then there are only 2 practical reasons not to do this:

1. To avoid the chance that you are wrong, and one of the libraries you use actually does malfunction when multiple elements have the same ID.
2. To maintain forward-compatibility of your website/application with libraries or services (or developers!) you may encounter in the future, that do malfunction when multiple elements have the same ID - which is a reasonable possibility since this is not, technically, valid HTML.

The power is yours!

Even if the elements are of different types it can cause you some serious problems...

Suppose you have 3 buttons with the same id:

<button id="myid" data-mydata="this is button 1">button 1</button>
<button id="myid" data-mydata="this is button 2">button 2</button>
<button id="myid" data-mydata="this is button 3">button 3</button>

Now you setup some jQuery code to do something when myid buttons are clicked:

$(document).ready(function ()
{
    $("#myid").click(function ()
    {
        var buttonData = $(this).data("mydata");

        // Call interesting function...
        interestingFunction();

        $('form').trigger('submit');
    });
});

What would you expect? That every button clicked would execute the click event handler setup with jQuery. Unfortunately it won't happen. ONLY the 1st button calls the click handler. The other 2 when clicked do nothing. It is as if they weren't buttons at all!

So always assign different IDs to HTML elements. This will get you covered against strange things. :)

<button id="button1" class="mybtn" data-mydata="this is button 1">button 1</button>
<button id="button2" class="mybtn" data-mydata="this is button 2">button 2</button>
<button id="button3" class="mybtn" data-mydata="this is button 3">button 3</button>

Now if you want the click event handler to run when any of the buttons get clicked it will work perfectly if you change the selector in the jQuery code to use the CSS class applied to them like this:

$(document).ready(function ()
{
    $(".mybtn").click(function ()
    {
        var buttonData = $(this).data("mydata");

        // Call interesting function...
        interstingFunction();

        $('form').trigger('submit');
    });
});

No. two elements with the same id are not valid. IDs are unique, if you wish to do something like that, use a class. Don't forget that elements can have multiple classes by using a space as a delimeter:

<div class="myclass sexy"></div>