In the past, it was believed that bilingual babies didn't realize they were hearing 2 languages. It was also assumed that they would develop a strange language merging their 2 languages.
Today, we know that bilingual babies are able to distinguish their languages as early as a few months of life. We also know that they will develop two separate language systems.
Babies just a few months old are already sensitive to differences in rhythm and sound; among other things, this is what enables them to distinguish their languages. This doesn't mean that a baby already understands that Mommy speaks a language called English and Daddy speaks a language called German. It simply means that a bilingual baby hears a difference between Mommy's sentences and Daddy's sentences. He/She's able to recognize his/her languages among others and between them.
Some sources to go further:
Hoff & Core (2015). What Clinicians Need to Know about Bilingual Development.
Byers-Heilein & Lew-Williams (2013). Bilingualism in the Early Years: What the Science Says.