No, because, it is the subject of the sentence. Is 'that' my chair. Since you are already mentioning it as a subject of the sentence, den/meinen/seinen do not appear here. You could use meinen in this case for example: Er hat meinen Stuhl. Here 'Er' is the subject. Hope you got it! :)
"That" and "it" are similar, but they are used to mean different things. "That" is used to point at an object, often showing the object's location or identity. "It" is used after the identity of the object has been communicated. Example:
"I'd like to buy that. Be sure to put it in a paper bag, not a plastic one."
As you can see, "it" refers back to something already mentioned, but "that" identifies the object being mentioned and introduces it.
In my understanding, the DAS here is not the indicator of gender - think of it as DAS ist mein Stuhl. DAS is a pronoun parallel to THAT in English. The gender of the noun is related to MEIN. Mein is masculine here. For example, we could have said: Das ist mein Stuhl, Das ist mein Sofa (Sofa is neutral - das Sofa), Das ist meine Liste (list is feminine). I am not sure if my explanation is clear but DAS in that case is gender neutral. Best wishes, Marina