From everything I've read, that's basically it. Babies get their linguistic start by just making babbling sounds "ba", "ma", "da", etc, because they are easy to produce. We as adults attribute meaning to them, that meaning is reinforced for the infant, and then it is accepted.

What I was curious about was why so many unrelated languages seem to use the "ma" sound for mother. If babies make random sounds and we assign the meaning to them, how did unrelated cultures agree on what each sound meant?

There's a section in this article that includes a suggestion of an explanation:

https://en.wikipedia.org/wiki/Mama_and_papa

Linguist Roman Jakobson hypothesized that the nasal sound in "mama" comes from the nasal murmur that babies produce when breastfeeding