In the scenario that I have a binary classification problem, and use a binary classifier to train and test my model, assuming everything else is constant, would using a multi-class classifier with 2 labels (one vs. all) produce the same results? (I know I should use the binary classifier, but I'm simply curious).
"Binary classification" is simply multi-class classification with 2 labels. However, several classification algorithms are designed specifically for the 2-class problem, where the response is modeled as the outcome of a Bernouilli trial.
Of course, this can be generalized to the n-dimensional setting by modeling as the outcome of a multinomial sample. For instance logistic regression can be generalized to multinomial logistic regression.
Some models, such as decision trees and ensembles of trees, can take a different form however when applied to n-class classification.
So the answer is highly dependent on the implementation of the algorithm you're using. But the baseline answer is that any multiclass classification algorithms can be used for 2-class classification, and most 2-class classification algorithms can be generalized for multi-class classification.