Does “people of Israel” and “Hebrew slaves” mean the same thing?

Yes, more or less.

The expression “people of Israel” refers to the nation descended from Jacob (also called “Israel”).

The word “Hebrew” has a wider usage. For example, Abram is described as a “Hebrew” (Gen 14:13). It is sometimes suggested that “Hebrew” derives from the name “Eber”, who was an ancestor of Abram (Gen 11:16) – this might suggest there were more Hebrews than just the Israelites. However, from the book of Exodus onwards “Hebrews” seems to refer to just the people of Israel.