No; an inner product space and a metric space are not the same thing. An inner product can be used to define a metric, as linked to from that Wikipedia page. But in general there are metrics that do not come from inner products; in fact, a metric does not even have to be defined on a vector space. A good example would be the discrete metric (also on that page, I suspect), which can be defined for any set whatsoever.
Is this describing a metric space? https://en.wikipedia.org/wiki/Metric_space
No; an inner product space and a metric space are not the same thing. An inner product can be used to define a metric, as linked to from that Wikipedia page. But in general there are metrics that do not come from inner products; in fact, a metric does not even have to be defined on a vector space. A good example would be the discrete metric (also on that page, I suspect), which can be defined for any set whatsoever.