Wiktionary isn't certain whether the Chinese 蘑菇 (mógū) is a loanword from Mongolian мөөг (möög) or vice-versa. Both mean "mushroom" and are pronounced very similarly so there's clearly some relationship, but I don't know which came first. I can't find any reliable English-language source that addresses this, and my Chinese level is insufficient to research in Chinese-language sources. It seems more plausible to me that the Mongolian word comes from Chinese, as many Mongolian vegetable words are loanwords from Chinese, but I don't have any resource to support that.
Thanks to @水巷孑蠻, I've updated this answer.
My Mongolian teacher thinks 蘑菇 is borrowed from мөөг.
After a brief search in Chinese, all I see is that 蘑菇 is borrowed from Mongolian. But those are just general articles, not scholarly credible.
Glyph Origin of 菇、菰、苽
- 菰 is old, at latest from 戰國 (Warring States). 菰 is zizania latifolia. Its meaning carries on to today.
- 苽 is also old, at latest from 漢 (Han). 苽 is basically the same as 菰.
- 菇 is young, and the earliest I can find is from 983AD 北宋.
Another usage is 慈菇/菰, which is arrowhead (sagittaria trifolia).
Glyph Origin of 蘑、藦
《詩·衛風》云：「芄蘭之支。」陸機云：「一名蘿藦。幽州人謂之雀瓢。」 《爾雅注疏 》 The quoted sentence is from 註 (晉), 276－324AD 蘿藦 is 芄蘭, which is a type of grass.
466-527AD（late 北魏）： 西南逕蘿蘑亭南 《水經注》 name of a place
963AD (北宋)： 又棣州合進蘿蘑子 《五代會要》 name of a place
1245AD （南宋）： 多生溪邊沙壤松土中，俗名麥丹蕈。未詳，味殊美絕，類北方藦菇，蕈品最優銀。 《菌譜》
after 1260AD： 更說髙麗生菜美，總輸山後蘑菰香。 《灤京雜詠》 灤京 is 元上都.
~1290AD： 北內送天花蘑菇 《武林舊事》
So 蘿藦 is the name of a type of grass. Later 蘿蘑 is used in names of places. I suspect it is a variant of 蘿藦. Until first in 南宋, 藦菇 is documented. 蘑/藦 doesn't mean anything on its own.
Coinage of 蘑（藦）菇（菰）
蘑菇 is not a compound word since 蘑（藦）is not a lexical morpheme and 菇（菰）referred to something other than 蘑菇.
蘑菇 is not a 連綿詞. 蘑 and 菇 are neither 雙聲 (same onset) or 疊韻 (same non-onset).
The unstable writing form hints that it may be transliterated.
The coinage of 蘑菇 is closely related to historical trade of calocybe gambosa (a type of specialty mushroom from Mongolia), not sure if other types of mushroom were involved, to other parts of China through 張家口 Zhangjiakou, thus another name 口蘑. But on the Mongolian steppe, there are other types of mushrooms. They are now all called xx мөөг. So мөөг is now used as a general term. Historical usage, however, is beyond my knowledge of Mongolian.
I did find juxtaposition of 香蕈 and 蘑菇 in 《西遊記》. So 蘑菇 upon coinage likely only refers to 口蘑. It's only later that it's extended to mean all mushroom.
What do ancient Chinese call mushroom?
In Classical Chinese mushroom is called 蕈 xun4 or 菌 jun4. This dates back to at least 2000 years ago. For example 香菇 is still called 香蕈 in Ming Dynasty.
I think the word 蘑菇 was coined during the 口蘑 trade by the Mongolians to denote the sound of мөөг.
The long-existing glyphs 蘑（藦）菇（菰）are used to transliterate the sound.
It is likely that 蘑菇 originally only refers to 口蘑, but later is extended to all types of mushrooms.
As for 菇（菰）, the lexical meaning of mushroom is shortened from the loan word 蘑（藦）菇（菰）. 蘑菇 replaced 蕈/菌 (especially 蕈 since 菌 still has limited use colloquially) in colloquial languages for mushrooms.
An answer on QUORA.com (by Tang Ek'iak on Sep 18, 2017):
"Chinese borrowed from Mongolian in Ming Dynasty.
But 菇 is not. Before this word, northern Chinese call mushroom 菌 and Southern Chinese call it 菰(菇). When people borrowed this word from Mongolian, they created 蘑 to combine with 菰/菇 into 蘑菰/蘑菇.
in 本草圖經, a Chinese materia medica work which was written in Northern Song Dynasty:
Today, some Chinese used this word for mushrooms in general, but some use it for one kind of mushroom."
Another source supporting this claim is "World Loadword Database (WORD)" - Article