Does he know immediately (from some internet / intranet access) which students in his classes are undergraduates, masters students, PhD students, non-degree students? Does he typically have access to his students' transcripts, thesis work (dates submitted / approved / topic), and knowledge of his students' academic / research advisors?
At my R1 state university in the U.S., I do have info on whether students in my classes are undergrads or grads or non-degree, which are auditing and which taking it for a grade.
I have no information on their transcripts, nor thesis work nor status, nor academic or research advisors.
For students that are my advisees, yes, I have such information. In fact, for PhD students or MS students whose committees I'm or have been on (that are still active...), I do also have such information. But not for any other students.
It varies widely.
First of all, federal privacy laws allow information to be shared with "school officials with legitimate educational interest", without needing student consent. Professors are generally presumed to fall into this category.
Basic information like the student's graduate/undergraduate status are usually listed in the course roster, and the professor very likely has seen that information (which is not to say he remembers it).
Beyond that, there is a range. Some institutions, based on their own interpretation of the law and risk/benefit analysis, choose as a matter of policy to make full transcript information available to all faculty members. Others choose to impose various restrictions on who has access to such data; e.g. academic advisors, department heads, administrative personnel, etc. I don't think they are generally obligated to tell students what those policies are, though if you ask around you can probably find out. So all you can really say in general is "maybe" and you should be prepared for either extreme.
In particular, as a student, you shouldn't assume this information can be kept secret from a professor. Even if she doesn't have direct electronic access to transcripts, she can likely get this information easily. In many cases, it would suffice to write an email to an appropriate person with access, often an administrative staffer, and give some plausible reason: "Dear Joe, could you please get me a copy of so-and-so's transcript? I need to check on blah blah blah." She could also just ask around: "Hey Alice, you are so-and-so's advisor, right? What are you two working on these days? How is it coming along?" Faculty usually don't hesitate to share such information with each other.
On the other hand, if there is some academic information about you that you think is important for the professor to know, you shouldn't assume that he will automatically know it. Even if he has direct access to those records, faculty usually wouldn't bother to read everyone's transcripts without some reason to do so. If there's some aspect of the course that is particularly relevant to your interests, based on your advisor and research topic, don't assume the professor knows / remembers that information: tell him!