I am wondering how effectively a stationary bicycle can be used at building lower-body strength vis-a-vis weight-based exercises like squats and lunges. For instance, if I put a stationary bicycle on a resistance setting of "15" (just a number, I understand that scales on different brands and models are different), I can't pedal it faster than 40 or 45 rpms or for longer than 3 or 4 minutes. When I am done, I can barely stand. Can I substitute this routine for a traditional set of squats?
Bikes are definitely not a substitute for squats
The motions are vaguely similar and workout out many of the same muscles but biking is cardio and squats are for improving strength. You might be the best biker in the word but that doesn't mean you can automatically come out and squat 400 pounds.
Another thing to keep in mind is the squat is a more full body motion. When you bike your quads and glutes are doing most of the work. In a full atg squat you bring in those your calves and your back.
To sum it up if you want strength do squats for cardio endurance do stationary biking.
Strength is built and demonstrated with intense exercise. In this context, "intense" means "it is increasingly difficult to do repeatedly". An intense squat for someone would be a weight they could only squat two or three times--or only once--before failing.
Intensity is inversely related to duration. If I can do something five hundred times, then by definition it's not intense (according to the strength-training definition of "intensity"). So no matter how wobbly your legs are after a 4-minute level-15 spin, the intensity of bike riding is fairly low. So that's one way in which squats, and strength training in general, are radically different from bike riding (unless the resistance is so high you can only rotate the pedals once).
The other major difference is that the movements are so radically different from each. Squatting is not the same as pedaling. There is some limited overlap, but different exercises are different.
Literally any physical activity can build strength. Riding a stationary bike on a high resistance has a strength component, but not much. If you want strong legs, or to gain the benefits of strength training, then do squats and strength training.