I'm trying to find a way to make dingbats appear exactly the same
You will never make fonts look exactly the same in all browsers, whether the characters in question are Dingbats or not.
For me, most of the characters on that page don't render in IE or WebKit. IE traditionally has poorer font fallback than average and Firefox typically better then average. The font that Firefox and Opera manage to choose to render the symbols for me is Meiryo (a Japanese font installed with Windows Vista and later). On IE and WebKit it falls back to the much more limited selection of symbols available in Arial, leaving most of the characters missing.
So for the best chance of rendering symbol characters how you want, do as you do for any other characters, and specify the font you want, eg. CSS font-family: Meiryo
. But of course anyone who doesn't have that font installed will get something different, and browser/OS settings may change how fonts are rendered in general.
The symbol characters from the Zapf Dingbats set are not safe to use on the web, as the basic default sets of fonts installed by operating systems do not typically include glyphs for most of them. (‘Wingdings’ on Windows does, but it's a legacy font with a custom mapping that puts the symbols on ASCII characters instead of where they should be in Unicode, so again it's not safe to use on the web.)
There are a few symbol characters that you can typically get away with using for commonly-available font sets, eg:
● ■ ☺ ☻ ♥ ♦ ♣ ♠ • ▲ ▼
others, I'd try to avoid.