You are right, "kobiety" is also the nominative of plural, but in this sentence you have to use the genitive. Hence "To jest kobieta. Nie widzę kobiety" = "This is a woman. I don't see a woman" and in plural: "To są kobiety. Nie widzę kobiet" = "These are women. I don't see women".
Well, I gave it some thought and read some grammar articles in Wikipedia and it seems to be like this:
the direct object is usually in the accusative or, in negated sentences, in genitive (but instrumental is also sometimes used);
dative, instrumental, locative are for indirect objects.
I guess this link https://en.wikipedia.org/wiki/Polish_grammar#Noun_syntax can be helpful. Some examples to show that the object can come in many different cases (it really just depends on the verb used):
"Nie widzę kobiety" = "I don't see a woman" (genitive)
"Przyglądam się kobiecie" = "I look at a woman" (dative)
"Widzę kobietę" = "I see a woman" (accusative)
"Rozmawiam z kobietą" = "I speak with a woman." (instrumental)
"Mówię o kobiecie" = "I talk about a woman" (locative)
There is no "tej" nor "tamtego", because there's neither "this" nor "that" in the English sentence.
"tamtego" is masculine anyway.
Well... you could say it's a suffix, I guess. "kobiety" is the Genitive form, Genitive is needed in this sentence.
OK, you have the word for "a woman": "kobieta". This is Nominative.
Then you want to say "I see a woman". "widzieć" (to see) takes Accusative. "Widzę kobietę".
Now, you negate it. "I don't see a woman". If a verb that needed Accusative gets negated, it takes Genitive instead. "Nie widzę kobiety".
Note that only Accusative changes when negated. Other cases stay the same.