Picture the one object in a house that everybody steps on and nobody looks at. It lies at the door. Its job is to take the dirt so the floor stays clean, so it is made of rough material and nobody minds it wearing out. It never moves and it never complains. Both meanings of this word live in that picture: the thing itself, and the person who has ended up in the same position.
From the word’s history
Wiktionary gives the word as simply door + mat: the mat that belongs at a door. The compound names the position, and the position is where both meanings come from.
Wiktionary 說這個字就是 door 加 mat：屬於門口的那塊墊子。這個組合命名的是位置，而兩個意思都是從這個位置長出來的。
Wiktionary, 'doormat' etymology ('From door + mat'), as supplied in this packet (source_ref kaikki:3f6f69a190036ae4:1171157274)
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