That is a halachic issue which you should address to a rov, but according to the Torah, no. Only if a child was conceived with someone else’s legal wife, is the child considered a halachic mamzer, according to my understanding of what i was taught.
Rabbeinu Gershom issued a takana that a man should not marry more than one wife concurrently, but that still does not make a child a mamzer, if he does. There might be another halachic designation (which i don’t know if there is or is not), but mamzeirus as it applies to your question specifically refers to the child born of adultery committed with a married woman. He might not be permitted to marry bigamously, but the woman, if she was single, is not committing adultery, so her child is not a mamzer. Two of our Avos had more than one wife at the same time, and they followed the Torah. The Torah speaks of the halachos of yerusha, as it relates to the children from two concurrent wives. Nowadays men don’t marry a woman while they are still married to another one (except in the rarest of situtations and with a heter meah rabbonim). But I imagine the halacha is still interpreted in the same way regarding mamzeirus, i.e. that the child is not one.