Answer:
No, poison ivy [Toxicodendron radicans] and poison oak [Toxicodendron diversilobum and Toxicodendron pubescens] aren't the same. What they are is related. They share the same genus [Toxicodendron], which is like people who share the same last name. But they aren't in the same species [radicans, pubescens, diversilobum], which is like having different first names.
Something else that they share is a lack of toxicity or poisonousness. Specifically, they both share the presence of urushiol. That oil causes a skin reaction because of irritation, not because of poisons or toxins.