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I intuitively think of the act of performing an arbitrarily large number of pushups as one requiring a considerable amount of strength. However, sometimes it is claimed that this is not the case.
Allowing that different people may have differing reasons for making this claim, i.e. the claim may have no single explanation, what are some possibilities of quantifiable definitions of strength on which this claim is being made?
"Strength" has a more specific meaning in sports and exercise science than it does in common parlance. Strength is the ability to exert a maximal force against a resistance [1].
A push-up is not a maximal effort for many people, and certainly not for someone doing more than a few in one set. A maximal effort similar to a push-up would be a maximum-weight attempt at a bench press. Therefore, using this domain-specific definition of the word strength, someone who can bench press 150kg once is stronger than someone whose best bench press is 100kg. This is true even if the 100kg-bench-presser can do more push-ups in one go than the 150kg-bench-presser!
Being able to do a lot of push-ups involves a degree of strength, but is more a test of muscular endurance. You can read more about these domain-specific terms in books about exercise science rather than general fitness.
[1] Definition from page 1 of Harvey Newton's Explosive Lifting for Sports, but it's defined similarly elsewhere.
We say this because when you keep doing pushups for 50+ repetitions, at a certain point, your muscles are really just adapting to this particular movement, and this particular weight, rather than being stimulated to grow. So someone who is able to do a huge number of pushups, might fail miserably if you asked them to do a few with say 10kg of added resistance.
It's not like there are no benefits though. While the strength gains are limited, improvements in stamina is the takeaway.
If you look at the bodies of seasoned marathon runners, you'll find that their muscles aren't particularly large. But they are superbly durable over time, and can keep going for hours on end without pause.
Then compare this to the body of a 100m dash sprinter, whose body is meant to perform explosively over short periods of time.
And then to a weightlifter, who is, instead of aiming for long periods of movement, aims for one single rep.
If you compare the bodies of these different types of athletes, you'll find how the difference is easily spotted with the naked eye.
This is the reason why our calves don't grow to the size of a volleyball when we climb mountains. They gain the strength needed to carry our bodies, but after that, they gain the stamina to keep doing it, rather than gaining the strength to carry ten humans.
Quantifiable differences though, isn't something I'm equipped to answer. This is more aimed at the question in the title.
You will build strength to a certain level (ie the level it takes to do the push up at your weight comfortably)
After that you are building stamina to continue the repetitions.
It would be like benching 60 kg a hundred times. It is then easy to do that weight but it's not preparing you for 120kg really
Waaaaay back when I was in the military, my maximum number of consecutive push ups was 84. I was only able to achieve around 56 push ups by concentrating on push ups alone. I was able to achieve that higher number through a combination of increased cycles of push ups and weight lifting.
The increased number of push ups increased my stamina more than strength. The weight lifting I did increased my strength and power, thus assisting with my push ups. I had increased my bench to around 260 pounds (117 kilograms) through dedicated weight lifting at the time I hit that maximum number of push ups.
Of course, my observations are based upon stamina being defined as duration of continued effort and strength being defined as power and maximum weight moved in only a few pushes/pulls.
Now, I'd be lucky to hit 20 push ups before collapsing....
You are mixing two assertions. One is that many pushups doesn't require a lot of strength and another is that it doesn't build strength. The first is false but the second is kind of true.
The stronger you are, the easier it gets and thus the more you can manage. If you can bench press 100 kg then it'll be very easy to rep out with just a 20 kg bar. However, if you're 1RM is 30 kg then you'll struggle going beyond 4-5 repetitions.
A high-repetition scheme does build some strength but is far inferior to low-rep/high-resistance in that regard:
If you cannot perform one pushup and after some training, you're able to do that, you have increased your strength. That's because you now have the ability to lift a large percentage of your body off the ground.
In the Strength training world, being able to perform between 5 - 8 repetitions of the same exercise (without reducing the load/intensity) is a sign of strength for the specific exercise.
On the other hand, muscular endurance is the ability to perform the same feat multiple times with little or no rest. This means that your body is adapting to the exercise and you're becoming better at it.
If you can do it once, your strength has increased; if done multiple times, endurance is kicking in. Usually, muscular hypertrophy is gained between 8 - 12 repetitions.
Repetitions higher than 15 are usually geared towards muscular endurance (meaning that one will be able to perform such exercise for a while before fatigue sets in).
None of these is good or bad; everything depends on the person's goals and intended benefits.
If you can do only 5 pushups, and you keep trying that single exercise until you can to 10, you will more than likely increase your muscle mass in the process.
If you can do 100 pushups, and you keep trying that single exercise until you can to 105, chances are that your muscle mass will not go up in that process. Instead, you will probably increase your ability to use oxygen and calories, while releasing waste like lactic acid. The challenge of the 105th push-up is not pure strength.
So if your goal is strength, when high repetition of pushups becomes easy, add more weight or switch to benching.
In order to perform a large amount of push ups, you would have to possess some strength to do so, but it is made up more of endurance than strength when it comes to high reps of anything. For example, people will say you need high endurance to run great distances, so why wouldn't it be the same for a great number of push ups, too. Don't get me wrong, I'm not saying push ups are cardio but if you are doing over 20 reps of them(or anything, for that matter) then they/it are/is not building your strength, you are doing endurance training, unless, as I stated before(in the comments), you have an insane workout like doing thousands a day it might build you a little bit of strength. But I wouldn't suggest it. You would hurt yourself(most likely).
