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?
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.]