Sometimes exercising less gives you better results. Every time I went for a bike ride during the summer I improved my speed or endurance until I could bike 37 miles at a fast pace. I was able to do that because I did it as a weekly workout. Doing push-ups every other day gives your muscles more time to recover. So it could give you better results than doing it as a daily workout. The best way to find out is to try both ways.
Pay attention to your progress and how you feel. Taking days off helps when your muscles need more time to recover. Some people should do them as a daily exercise. Others should do them every other day. I usually do them as a daily exercise but I take days off when my muscles need more time to recover.