Contrary to popular opinion, the plump, brown bear is not named after that other plump, brown thing which is flushed down toilets.
The word Pooh is a quite modern word. It is a shortened version of pooh pooh.
Pooh pooh was a common phrase used by children particularly during the early part of the 20th Century to describe something that was silly or annoying .
It comes from the plosive sound of P.
A plosive is a letter which allows air to flow freely from the vocal chords. Other examples are B, D, G, K and T.
In many cultures the P, B and D are some of the first sounds made by babies. Thus baba, dada and papa.
(The very earliest sounds in a child's life are normally the nasal sounds. These are N, M and Ng. These sounds concern the sound coming from the nose. They are the earliest sounds as they are connected to crying and sucking. Thus explaining: ma, mama, nana etc.)
A.A Milne describes his meaning of pooh at the beginning of Winnie-the-Pooh
"...and whenever a fly came and settled on his nose he had to blow it off. And I think - but I am not sure - that that is why he is always called Pooh."
The use of pooh to describe the brown things found inside toilet bowls became popular from about the 1950s. Whereas Winnie The Pooh was introduced to the public on Christmas Eve 1925, in the English newspaper The Evening News.