Has a Peculiar Taste?

A person can be said to have "peculiar taste" - it means that they like strange things! If you're talking about how something actually tastes (the flavour), then you say it has "a peculiar taste".

What is a peculiar taste?

strange or unusual, especially in a way that is unpleasant or makes you worried. a peculiar smell/taste. The meat tasted rather peculiar.

What is have a taste?

1. To have a preference, inclination, or desire for some particular kind of food or drink. I've always had a taste for red meat, so I don't think I could ever be a vegetarian. ... I've never had a taste for fighting of any kind, so I know I would not be suited to the army whatsoever.