Why the HTML entity doesn't work on the Windows Chrome?

This is primarily a font issue, though there is a nasty silent change in HTML specs involved.

Modern browsers interpret ⟨ and ⟩ as referring to U+27E8 MATHEMATICAL LEFT ANGLE BRACKET “⟨” and U+27E9 MATHEMATICAL RIGHT ANGLE BRACKET “⟩”, informally known as “bra” and “ket”. This interpretation is being made official in the named character references section of HTML5.

These characters are adequate for use in many mathematical notations, and the ISO 80000-2 standard explicitly specifies that they are used e.g. for certain scalar product notations. But support to them in fonts is rather limited. In old Windows systems, no font contains them. In newer Windows systems, from Windows Vista onwards, Cambria Math should be available. It is possible that you have been testing on an old Windows version, but it is also possible that Chrome is unable to find the right font. To give it a helping hand, use a CSS rule that suggests that font, e.g. with the attribute

style="font-family: Cambria Math"

You might consider adding some other fonts to the list, using fonts that are known to contain the characters. See my Guide to using special characters in HTML.

The nasty change is that in HTML 4.01, in the entities section, ⟨ and ⟩ are defined as referring to U+2329 LEFT-POINTING ANGLE BRACKET “〈” and U+232A RIGHT-POINTING ANGLE BRACKET “〉”. They are logically less satisfactory (and deprecated by the Unicode Standard), but they have somewhat wider font support.

So in addition to declaring fonts that contain the characters you use, you need to decide which pair of these characters you use or whether you use something else; it's a complicated question. If you use them, it is best to use them as such (in a UTF-8 encoded HTML document) or using numeric character references such as ⟨. The reason is that ⟨ and ⟩ should not be expected to work consistently; they probably work the HTML5 way in all modern browsers, but there is hardly any reason to take the risk, when you can unambiguously indicate the characters you want.

That particular character is simply a unicode codepoint which is an arbitrary number. There are a lot of unicode codepoints that do not have an 'official' symbol. Even if they do have a symbol, it is not necessarily the case that your font has a symbol for that codepoint. If you choose a different font, you may end up with a different symbol.

I looked at the CSS for the page and it shows this character displaying in Arial (plus a bunch of other fonts that do not matter). Windows comes with Arial so it should always pick up that font first. It looks like Arial does not have a symbol for that unicode codepoint. Anytime you do not have a glyph for a codepoint, it puts in some form of a box indicating there is no glyph