Learn Polish: Ona, podobnie jak matka, jest bardzo pracowita. - She, like her mother, is very hard-working.

Both translate to "like", and using just jak (Ona, jak matka...) is perfectly fine. However, podobnie jak literally means "similarly as" and is a very common set phrase used to strongly emphasize a shared characteristic. It translates best to "just like" or "similarly to."

What case is matka here? Doesn't a preposition usually change the case of the noun?

Matka is in the nominative case (the basic dictionary form). Jak isn't a typical preposition that forces a specific case; it acts as a conjunction here. When comparing two things using jak, the second noun matches the case of the first. Because Ona is the subject (nominative), matka is also nominative.

Why doesn't the sentence say jej matka (her mother)?

In Polish, possessive pronouns like jej (her), mój (my), or twój (your) are frequently dropped when the context makes it obvious whose relative, item, or body part is being discussed. Adding jej wouldn't be wrong, but leaving it out sounds perfectly natural and less repetitive.

Why is pracowita in the nominative case? I thought words after jest (is) take the instrumental case.