As of now as far as the tech aspect is concerned, answers to your 5 points are:
Do assistants have personality traits?
Most have a basic persona attached to them, which is noticeable with either "joke" queries or with predictable queries such as "How's my schedule looking?". In either case, for now, there's no algorithm to create new responses, but instead, there's a hefty bunch of thoughtfully written templates which are then used to form a response.
Do they learn from every interaction?
But that tendency to learn is more inclined towards looking at the success ratio for completing user queries adequately, and what the user is likely to ask for.
Can they modulate the voice?
Not in the way we do. There are multiple voice groups, accents and even intonations but those are different. Google has been working on Expressive Speech Synthesis which aims to get better at the natural flow of speaking and has been quite successful (technically), but there are a ways to go.
Can they have moods?
They are meant to be dependable and alert at all times, which wouldn't allow for inconsistent behaviour brought upon by "moods" as a concept.
Depending on the owner and the situation, can the degree of formalization of the relation change?
Again, there has been some noise about being able to customise the actions with respect to identified individuals around the primary user, but I can't find a source right now to back that up.