Are push ups an anaerobic exercise?

Typically yes.

Exclusively, no.

Most resistance training (moving your bodyweight is a resistance) is anaerobic because you can't replace energy fast enough with oxygen (aerobic) to meet demand.

Meaning most resistance training exercise will tap out around the 90 second mark before you can't complete more reps with good form.

You can train to push beyond this provided the resistance is low enough though. At which point it can become more aerobic than anaerobic but the resistance would typically need to be less than 40% (probably more like 20%) of your potential max load tolerance for that to occur, or you will have had to train your mitochondria to be more active during the push up by training it quite often.

People view energy systems as either or, which is a gross simplification.

Your aerobic system is always on in the background even when doing predominantly anaerobic exercises. The three energy systems work in tandem with one another, not exclusively of one another.

So after you do a set of anaerobic pushups for 60–90 seconds, your aerobic energy system is what helps you recover so you can do another set of 60–90 seconds.

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