bathing gown is not familiar to me. It's not in the OALD, CALD, AHDE or Merriam Webster.

A quick search suggests that it may be an archaic term to describe a garment worn for bathing, that is, a gown worn in the water, which would mean it is not equivalent to a bathrobe.

http://marthawashington.us/items/show/48

If you want Bademantel, then bathrobe is the word you should use, at least in the US. If I'm not mistaken, the same garment is called a dressing gown in BE.

in which case the pair "bathing gown = Bademantel" might be a candidate for "Wrong Entry"?
Siehe Wörterbuch: Bademantel

I can confirm that in BE this is called a "dressing gown".

I agree; 'bathing gown' looks like it was made up by Doris to catch plagiarists.

In my family at least, the normal forms are simply robe (Bademantel) and gown ([Damen]Nachthemd). The more traditional long forms bathrobe and nightgown are understandable, but I would be surprised to see them still used in conversation as opposed to in print.

I (AE) believe that, regarding male attire, there is the following difference between a bathrobe and a dressing gown.

The bathrobe (or just "robe") is much more common at all social/income levels. It is often of terry cloth, flannel or other water-absorbent material. It is worn not only to and from the bath, but also over pajamas late at night or in the morning (perhaps even at breakfast).

In contrast, the dressing gown is worn indoors in the evening over shirt and trousers -- that is, instead of a sport jacket or suit coat. It is usually made of elegant material, perhaps quilted, perhaps with satin lapels. The dressing gown is worn only in upper-income households; one thinks of movie stars in the 1930s.

Regarding the OP, I suggest that "bathing gown" is simply a mistake, and that "bathrobe vs. dressing gown" is the real dichotomy.

I'm surprised to hear hm--us say that nightgown and bathrobe are old-fashioned. We say both. Perhaps that's a regional difference. I suppose I might say robe for bathrobe too, but I definitely don't say gown by itself unless I am talking about a fancy dress, and then it would sound dated.

'bathrobe' is not uncommon in BE, but 'dressing gown' is more usual. In his diary, Samuel Pepys called it a 'nightgown', one of a number of false friends to which the editors of the definitive edition draw attention. 'Nightgown' is a word I haven't actually heard for ages, and 'bathing gown' a term I have never heard, ever, though it has plenty of google hits from apparently native sources.

I partly agree and partly disagree with ion1122’s exegesis (#4).

The bathrobe (or just "robe") ... is often of terry cloth, flannel or other water-absorbent material. 
That ties in with my own (Australian) experience.

It is worn not only to and from the bath, but also over pajamas late at night or in the morning (perhaps even at breakfast).
To me, that is a dressing-gown – which doesn’t have to be highly water-absorbent. (Someone from Sydney once told me this was a shaving coat – are any LEOnids familiar with this expression?)

In contrast, the dressing gown is worn indoors in the evening over shirt and trousers -- that is, instead of a sport jacket or suit coat. It is usually made of elegant material, perhaps quilted, perhaps with satin lapels. The dressing gown is worn only in upper-income households; one thinks of movie stars in the 1930s.
Edith wouldn't know what to call that, because we don't have them in Australia. (But to us, it's definitely not a dressing-gown.)