The only way in which that strikes me as "weird" at all is that the second person responded in a different style than the person they are responding. That is, there's nothing inherently weird or awkward about the period. Imagine, for example, this instead:
> I'll be home late tonight.
>> Okay.
I don't think anybody would bat an eyelash at either part of this exchange, because they "match" in style.
> I'll be home late tonight.
>> Okay.
I don't think anybody would bat an eyelash at either part of this exchange, because they "match" in style.