The same thing happens in Spanish ("subir arriba" and "bajar abajo"). It became a serious pet peeve after someone pointed it out to me, until I noticed that people only make this "mistake" in either very noisy places or very time-sensitive situations. It's more a desired redundancy (even if not conscious) than the accidental redundancy in the case of au jour d'aujourd'hui.
Writing about this reminded me of how some people will try and make conlangs that compress as much information into as few sounds as possible, only to render them unusable - whereas within a rounding error all natural languages have the exact same information density and Signal To Noise ratio.
Edits: grammar, formatting.
"I climb up" might give a somewhat better impression of how "je monte en haut" sounds in French, although I suppose that you can very much "climb down" in English so it's not a perfect match.
Wiktionary[1] says that the expression has existed at least since the 18th century, adding this interesting note:
> À noter aussi que durant ce siècle, la notion de « lumière » était une représentation métaphorique de la « connaissance » (cf. : siècle des lumières). Cette expression peut donc aussi se traduire de la façon suivante : « à la connaissance d’aujourd’hui » ou « au savoir d’aujourd’hui ».
Roughly translated it says that light could be used metaphorically to imply knowledge, so "au jour d'aujourd'hui" could be parsed like "to the light of today" and from there "to the knowledge of today". Now whether that's a justification after the fact or the real origin of the expression, who knows...
[1] https://fr.wiktionary.org/wiki/au_jour_d%E2%80%99aujourd%E2%...