No. The country issuing the passport is the passport bearer's country of citizenship. The passport was issued under the authority of that country, and it certifies that the country recognizes the bearer as a national of that country. The bearer of the passport is therefore not stateless.
The place of issue of a passport is just that: the location of the office that issued the passport. It is quite common for foreign diplomatic missions to issue passports to their citizens abroad, and when that happens, the passport's place of issue will usually reflect the fact that the passport was issued in a country other than the one on whose authority it was issued.
To say this more concretely: A passport issued by the embassy of Aliceland located in the country of Bobistan will often say that it was issued in Bobistan. But it was still issued by Aliceland, and the person named in the passport is still a national of Aliceland.
See also Which country issued my passport if I received it at a consulate abroad?