eigenartig
strange
Odd, peculiar, or weird; deviating from the norm and therefore causing surprise or puzzlement.
Er hat einen sehr eigenartigen Humor, den nicht jeder versteht.
He has a very strange sense of humor that not everyone understands.

strange / characteristic
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strange
Odd, peculiar, or weird; deviating from the norm and therefore causing surprise or puzzlement.
Er hat einen sehr eigenartigen Humor, den nicht jeder versteht.
He has a very strange sense of humor that not everyone understands.

The word is made up of 'eigen' (own) and 'Art' (kind/type). It describes something that has its 'own way' or 'own kind.' This explains both meanings: something with its own way can be strange (m01) or characteristic (m02).
Sein Kleidungsstil ist ziemlich eigenartig.
His style of dress is quite strange.
The meaning 'seltsam' (strange) is more common in everyday language and often refers to people, behavior, or situations. The meaning 'charakteristisch' (characteristic) is more formal and is often used in descriptive texts, e.g. about art or science.
Das Essen hatte einen eigenartigen Nachgeschmack.
The food had a strange aftertaste.
In the meaning 'seltsam' (strange), the adjective is often connected with sensory perceptions or people. Typical phrases are 'ein eigenartiger Geruch' (a strange smell), 'ein eigenartiges Gefühl' (a strange feeling), or 'ein eigenartiger Mensch' (a strange person).
Als ich den Keller betrat, hatte ich ein eigenartiges Gefühl.
When I entered the basement, I had a strange feeling.
In the meaning 'charakteristisch' (characteristic), 'eigenartig' (characteristic) often appears with a dative object that indicates for whom or what something is typical. The structure is: 'etwas ist [Dativ] eigenartig' (something is characteristic of [dative]).
Die besondere Farbgebung ist diesem Maler eigenartig.
The special use of color is characteristic of this painter.
own / peculiar
'eigen' and 'eigenartig' look similar and can both describe something special or unusual. In everyday language, 'eigenartig' usually means 'strange' or 'odd'. But it can also mean a special, characteristic feature. 'eigen' has several important uses: it can express possession, as in 'mein eigenes Zimmer' (my own room), or describe a special kind or quality. That is why the words are easy to confuse. For possession, use only 'eigen'. When something seems weird, 'eigenartig' is usually the clearer choice.

Nach der Arbeit hatte sie ein eigenartiges Gefühl im Bauch.
After work, she had a strange feeling in her stomach.
Auf Reisen nehmen wir oft unser eigenes Essen mit.
When we travel, we often take our own food with us.