Can we increase muscular strength by lifting weights with more repetitions until fatigue, similar to low repetition with high resistant training?
I'm going to have to assume that buy strength you purely mean the ability for the musculature system to generate force and not hypertrophy. If not then it will change my answer slightly.
So the two methods we are talking about here is the maximal effort method and the repeated effort method. The maximal effort method involves using a weight thats 90% or more of your 1RM usually for 3 or fewer reps per set. The repeated effort method consists in using between 70-90% of your 1rm and sets are done until failure, not 1 or 2 reps before you literally squat until you go down and don't come back up again (not for the faint-hearted or novices) I wouldn't use this method with beginners as they often suffer for fatigue induce technical breakdown and when this happens you increase the risk of injury exponentially. That aside the method is advantageous but should be exercised with caution.
Strength isn't just a product of muscle but the ability of your CNS to fire the maximal amount of motor units in the correct sequence. You CNS is like a conductor controlling the timing of the orchestra to produce a symphony of force. Don't get me wrong; you still need muscle and increasing its mass especially myofibril hypertrophy will increase the number of actin and myosin filaments that the motor units control improving the force output of the muscle. But if your body cannot use the maximal amount of motor units in the correct sequence, it will not be strong and having big muscles will be useless.
The maximal effort method is considered superior for improving both intramuscular and intermuscular coordination. The muscles and CNS adapt to the load placed upon them. This method fires the maximal number of motor units with optimal discharge frequency; thus this method should be used to bring about the most significant strength increments.
The repeated effort method can also be used to fatigue the maximal number motor units, but because the mechanical loading is submaximal (less than 90% of 1RM), your body will be required to go to failure to ensure all motor units are fatigued. During the first few reps, the body will recruit the slow, fatigue-resistant motor units first only when these are fully fatigued with the body recruit the high threshold fast motor units. So if reps aren't done until the point of failure, the bigger motor units will not be recruited and therefore not trained, and maximal strength increments will be slower. This method is good to increase muscular hypertrophy.
For a complete training program, both methods should be used at different periods of training to ensure the full development of the body.