Can doing high repetitions of push ups increase my strength?

High repetitions of push ups will improve strength to a certain degree, but they will mostly increase your muscular endurance. If your personal goal was hypertrophy (build muscle) or maximal strength (lift heavy loads) you would want to use exercises that allow you to lift heavier loads. Bench presses, chest press machines, and incline bench presses are exercises that are done in more stable environments and allow you to lift heavier loads, which is a requirement for muscle building and maximal strength. With a push up, you are very limited in your abilities to increase the load you're pushing against as you're load is essentially your body weight. You could potentially increase the load by using modalities such as a weighted vest, but those may not be readily available or practical.

 

High repetitions of push up can help you with endurance strength, but as with any strength training exercise, you need to increase the intensity to get stronger .

The trick with body weight exercises is how to increase the intensity! There are many ways to increase intensity in your push up practice like:

Changing Tempo: Prolonging the negative movement can add intensity to your pushups. This means lowering your body to the floor as slowly as possible. You can lift your body in one second and then lower the body in 3-5 seconds. Another way to add intensity to your pushups is to pause intentionally for three seconds at the middle of your rise.

Proprioceptive challenge: Another excellent way to increase intensity is to push up with one leg lifted or positioning both legs on a bench or a higher surface. Even more intensity can be applied by positioning one leg on a bench and the other leg lifted in the air. Although it's necessary to apply intensity in your practice to get stronger; it's also advised to use these techniques gradually and to step up only when your body has successfully adapted to the current phase.

To a certain degree yes. However, doing high repetitions of push-ups will increase the endurance in the muscle or the duration it can perform work, but not the ability to exert force (or strength). Eventually, one has to overload the muscle by increasing the weight and intensity of the exercise. This can be done by incorporating other types of exercises such as dumbbell chest presses or bench presses which the measured weight can be added to the workout load.

Doing high repetitions for push up will help increase your muscle endurance but not strength.  In the beginning you may see an increase in strength but your body will not get stronger unless you are adding more weight to you.  As most people are trying to lose weight and bodyfat you will not see an increase in strength over time since more than likely you will be lighter.  Can you however make it more challenging by using different pieces of equipment like the TRX or weight vest.