How common is mandatory attendance-taking in college courses in US? A recent SE Academia question asked about how best to implement a university's requirement that the instructor must keep attendance records for all students. The top-voted answer there was to simply refuse to follow the university's policy, specifically, "Allow all students in class and do not monitor attendance." The justification and comments indicate that this is such an unusual and egregious demand at a university that it should be refused on its face. 

Coincidentally, around the same time this topic independently came up at my own institution (large college in northeast U.S.), where mandatory attendance records, recorded by instructors and collected by the college, have been the process for many years, since before I arrived. 

When this discussion came up in my department I cited the SE question linked above and its top-voted answer. However, it seems that almost all of my colleagues (~40 faculty members) simply disbelieved what I was saying. Their sense was that mandatory attendance-record-keeping was standard and customary at any academic institution in their experience. At least one comment was made asserting that other institutions forgoing attendance records would be a sign of lowering academic standards. 

So: What is an evidence-based estimate for the proportion of U.S. college classes that have a mandatory attendance-record-keeping policy? A proper answer for this would require some kind of reference for the evidence of the estimate. The estimate could be at the level of the university, college, division, department, course, or something else, depending on available data. Additional points if a historical profile can be given, i.e., confirmation on whether attendance requirements have been growing more or less common in recent decades.
In the US there are actually some federal rules governing attendance keeping. Colleges must account somehow for students receiving federal aid, for example. That is a lot of students, as it likely includes those with government backed loans. While the rule doesn't seem to require that instructors keep a formal log, they are required to know who is attending and who is not. 

For example, Here is a sample page from a large US state university: https://uc.uiowa.edu/faculty/teaching-instructional-policies/student-attendance-absences.

Some states will, themselves, impose additional rules, as will some individual privately funded colleges and universities. 

There are a variety of reasons for such rules, including liability. If a student "disappears", especially one who is not technically an adult, the powers that be will want to know. 

Beyond such rules, some instructors count attendance in the grading rubric, but I don't think that is especially common post-secondary. But non-attendance can be a symptom of other problems and so instructors are expected in general to know who is there and who is not - especially when missing class or other activities is repeated frequently. 

So, while I don't know what percentage of colleges require this, my guess is that it is fairly high, based on the need to keep records. But it is also my guess that the rules aren't strictly honored in a lot of places.