“So X, as we said in the previous video, is an X dimensional vector, given that the parameters of logistic regression will be W which is also an X dimensional vector, together with b which is just a real number”

b is the bias term also called as the intercept. It’s a single real number, not a vector. Each weight wi is a real number, and is associated with one of the input features xi.

The logic is the same with the line equation y = mx + b.
Here the intercept is the value of y when x = 0.

In logistic regression the bias term is the “default” log-odds for the case that all predictors equal 0 (or equal to a reference value for categorical predictors).