Push up depths vary for every kind of person, goal, or even style of pushup. One of the great things about push ups is that there are numerous variations in which you can tweak the movement and get something a little different out of it.
For normal push ups, the optimal depth is when your elbows go just past 90 degrees. That is a little different for everybody but a good standard depth. You want to lower until your arms are bent JUST past 90 degrees because will keep you in a good shoulder position as well. Once you start to go lower than that (which you can) the risk of irritating your shoulder due to glenohumeral anterior glide is much higher.
You build strength at many ranges of motion but is it ideal? I can get really strong at the top of a push up by doing weighted quarter reps… is that ideal though? Probably not. For many people that can’t perform full range pushups I will recommend half reps at times but I will usually favor in full range of motion push ups BUT just doing the lower portion (Eccentrics). So instead of just giving a client normal push ups as an exercise, I can modify it and prescribe them elevated push ups, modified push ups (from knees), or we can do normal full range push ups but just the lower portion. This is ideal for building strength because you are taking the muscles through the full range, it is a lot more difficult to strain the right muscles without get to a good range of motion. For me there is no minimum depth, you can either hit the correct depth or not. If you can’t, no problem, time to make a change.
Getting better push up depth is best by doing other variations of push ups. I recommend elevated push ups where your hands are on a bench or a coffee table and modified pushups where your knees are on the ground. These two variations change the lever point and angles to where it makes the movement a little easier. Also, doing tempo work will really help you gain strength in the movement. Just because you can’t do the full range of motion doesn’t mean you can’t work on other aspects. I would recommend push up iso holds at at the bottom of the movement and eccentric lowering with assistance up.
In general, push ups are difficult for most people. They are difficult and they are difficult to do correct. What makes them the more difficult than a lot of other exercises is the core component that is involved. My best advice to get better at doing push ups is to 1) get a stronger core. Do more planks and work on having perfect positioning. 2) Train the tempos (iso and eccentrics) and 3) if you don’t have a coach, video yourself and make sure that you are keeping your core engaged, go low enough, and have overall good form.