How to find out which font contains a certain special character

Are you sure the characters are in the font?

Browsers search other fonts to find missing characters - therefor you cant use browsers to test font contents.

You can't trust applications either - here I am using an Arabic font in Notepad to show Chinese characters that I strongly suspect are not in the Arabic font. It seems to me the application (or the underlying Windows libraries) are searching other fonts to provide the missing character.

If you need an independent corroboration, you could try FontForge or another third-party font tool.

my browser should automatically detect that the glyphs are not in the font and find an appropriate match

Unfortunately, this assumption on OP is only true for Firefox. Firefox uses fontconfig which does per-glyph font substitution; appropriate font is used if it contains concerned character.

On comparison between IE and Chrome, IE is slightly better in that it provides more extensive fallback on per-script basis and provide UI settings for fonts, though that didn't cover Egyptian Hieroglyphs. The font fallback mechanism for Uniscribe / DirectWrite, the components mainly for complex text layout, is described in this MSDN article.

Though Chrome finally supports per-script font fallback on 2012, users have to install extra extension to tweak them — one such extension is Advanced Font Settings. However telling all site visitors using Chrome to install extension isn't quite feasible.

About the original problem, one might want to investigate on the direction of font fallback mechanism, probably some other bad behaving font overrides Noto font, claiming to support those glyphs while in fact it doesn't. A tedious method is to remove extra fonts on system and re-add them one by one until the culprit is found.