I don't think there is a canonical "right" answer to this question as there is no universal formulation of the terminology, so let me try and pick apart a few of the things you mention, and how I understand their use within the field.
The term "bit" can mean a couple of slightly different things. It can refer to how data is actually stored. This is where the quantum analogue, the qubit, comes in. Bits can also be used to measure the amount of information in something. Essentially, this is a measure of "if I had to write down this information, how many bits would I need to store the data". However, here, there is no direct quantum analogue. If you have quantum data, the only way we can get information about it is to measure, at which point, the output is classical, i.e. bits. So the measure of information is still essentially the same - bits not qubits. (Perhaps another way of putting it is that information measures your "surprise" of getting a particular outcome, whether that's tossing a coin, rolling a die, or measuring a qubit.)
Are Shannon information and what everyone here is calling "classical information" the same
If they're talking about a specific calculation of the information, then yes. "classical information" might instead be referring to the broader theory. It depends on context.
How does quantum information relate to, diverge from or reduce to Shannon information, which used log probabilities?
What people are more often interested in are averaged quantities such as entropies, conditional entropies and mutual information. These have direct quantum analogues, calculated based on density matrices of qubits rather than classical probability distributions. The density matrix still represents a probability distribution but, rather than using a single fixed basis (i.e. the "0" and "1" of a bit), there's a continuous range of possibilities, which change the actual calculations a bit.
What is the analogue of Shannon's information theory for quantum information? I have more often heard the term "quantum theory of information", but not sure if this exactly corresponds to what I have in mind as to what "quantum information theory" would mean in the Shannon sense.
The term "quantum information theory" tends to get used in two different contexts. One is extremely broad, covering the whole field of quantum information processing, computing etc. The other is much more specific and really does refer to the study of these (quantum) entropies, channel capacities and the like.