Is classical liberalism the same political philosophy as the liberalism today?

No. In fact, "classical liberalism" is almost the exact opposite of the "liberal" positions of the modern day.

The reason is simple; "liberal" is good. Everybody wants their views to be considered "liberal", and so the very Word "liberal" becomes distorted.

At its root, "liberal" refers to someone who supports liberty and freedom for each individual. Modern "liberals" are more concerned with controlling the behaviors of individuals so that everyone behaves the same way.