I think they are suited to different fitness goals.
Since your goal is to increase the number of consecutive pushups, I would recommend option 2. This
- removes the fear of performing pushups over time (personal experience).
- conditions your body to performing pushups.
- allows your body to maintain relatively high metabolism throughout the day (if done at strategic times of the day).
- allows your body to maintain some form of agility/springiness throughout the day, which might be good for your daily activities or warmups to your exercise regimen.
That being said, I don't think that both options will achieve your goal without this caveat: increasing the number of reps periodically.
If you perform 5 pushups ten times a day, your body will adapt to only performing 5 pushups. If you perform 5 pushups once every other day (or everyday), your body will adapt to just that repetition.
But if you periodically increase the repetition (which is basically the concept behind most successful exercise programs), you'll be able to perform as many reps as you desire over time.
Spreading 100 repetitions into 4 sets throughout the day will burn more calories over time than performing them all at once (personal experience supports this, would be glad if someone can validate or deny the claim).
Also, spreading it throughout the day will make it easier for you to have an adequate exercise regimen. It might even lead you into performing more exercises within that time-frame.
This has been my experience so far. Hopeful it's the same with most people.