Html duplicated ID

Even though is is wrong there is nothing wrong with the selector in jQuery

$('#Container1 #TextBox1').val(1)
$('#Container2 #TextBox1').val(2)

Demo: Fiddle

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A better choice will be use attribute selector

$('#Container1 input[id="TextBox1"]').val(1)
$('#Container2 input[id="TextBox1"]').val(2)

Demo: Fiddle

I do understand that "Id" should be unique, however, with the outest with different "id", can it help in solving the problem?

No. Having non-unique element ids will not work. Not consistently, in any case (different browsers and frameworks may handle this case differently).

Any experts out there can give me some good advises?

Prefer using class over using id, particularly for any component which may be reused multiple times on a page.

Set ids against the containing elements themselves instead of the internal component elements, and revise your jQuery selectors accordingly. Or alternately implement your component such that it takes a 'namespace' parameter/attribute when used, and prefix each classname with the namespace inside of your component (this approach works particularly well when creating custom JSP tags).

You absolutely should not have duplicate IDs. It may work*, but it is semantically incorrect and you should not do it

You should restructure your jQuery, however much that may stink. The best option would be to use a class, perhaps using the specific id of the parent to specify which one you want

Another less attractive but viable way would be to add a number or something to the end of the ID to make it unique then use jQuery to detect any elements with a specific part of an ID

* - As Arun describes jQuery will accept the selector, but it is NOT favorable because it is incorrect

I would suggest you to use class instead of id. Duplicate id's are not a good practice.