English can be a flexible language at times, believe it or not. Any of the following sentences is correct grammatically:
1) What kind of names do people in your culture give pets?
2) What kind of names do people in your culture give to pets?
3) I will give my daughter this book.
4) I will give to my daughter this book.
5) I will give this book to my daughter.
And there are probably many other combinations and permutations of word order that I could give you (or give to you), but I'll stop at five. As for the reason why your two sentences have to be worded differently, I'll leave that to the grammarians.