When something is called peculiar or having a peculiarity what does that mean? Does it just mean it has some specific features or does it mean that those specific features are unusual, not normally expected?
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Pecularity means "an odd or unusual feature or habit", but also "a characteristic or quality that is distinctive of a particular person or place".
Peculiar means "strange or odd", but also "unusual", or (in peculiar to) "belonging exclusively to".
Peculiar and having a peculiarity have both the meanings you supposed.
Informally, peculiar means also unwell: she felt a little peculiar for a while.
[Reference: the New Oxford American English.]