Yes, all plants put lots of protein in their seeds to give their offspring plenty of resources to get off to a good start. This does not mean that there are not major differences in protein content between the seeds of different types of plants. OP asked about legumes vs non legumes, not seeds vs other parts of plants.

Comparing protein content on a per cup of the food as it is eaten is going to give you a misleading answer to the question. We add water to black beans while cooking before we eat them, which increases the volume of the beans, so fewer beans and less of the protein they contain, fit in a cup after cooking than before. Eating a half cup of pistachios will leave you much more full than eating a cup of black beans. It is more meaningful to compare foods on a dry weight basis, so how much protein per 100g of dry peas, wheat, rice or nuts.

Peas and beans have 20-25g of protein (there are lots of different species, so some may be higher or lower), wheat is 8-15g (some may get higher, different types of wheat have different protein contents), rice is 10-12g. Tree nuts are in the 20g range, so close to legumes, but remember they are trees with huge root systems. It is not surprising their seeds are different from non legume annuals like rice and wheat.

Grains such as rice, wheat and corn are also seeds and they have less protein than legumes on a dry weight basis.

Edit to address whole plant differences: If you look at forage crops, legumes such as alfalfa have much higher protein content in the whole plant than grasses.

This is a good answer, but I have one minor change:

Legumes don't get the nitrogen for free. They trade for it by giving the bacteria sugars and a place to live. When legumes grow in soil with plenty of nitrogen, they stop trading with the bacteria (if you dig these plants up, you won't find as many nodules where the bacterialive on the roots). Still, legumes have much easier access to nitrogen than other plants.

All organisms need energy and they do this through chemical reactions called redox reactions. Redox is short for reduction (which means an atom/molecule gains electrons) - oxidation (which means an atom/molecule loses electrons). Some elements like oxygen are really hungry for electrons, while others (carbon and hydrogen) are relatively happy to give up electrons. When a redox reaction occurs that makes everyone happier, energy is released. Oxygen is important for life because it wants electrons more than anything else that is readily available. Because oxygen wants electrons so bad, more energy is released when it gets electrons. If a reaction gave electrons to sulfur rather than oxygen, less energy would be released.

A common redox reaction is respiration, where sugars (mostly carbon and hydrogen) react with oxygen (O2) to make water (H2O) and carbon dioxide (CO2). O2 oxygen is unhappy because there are two oxygen atoms stuck together, and each of them are trying to hog all of the electrons. Carbon and hydrogen don't pull on electrons very hard, so sugar essentially has an excess of electrons. When the redox reaction occurs and these atoms are rearranged, you end up with oxygen bonded to carbon or hydrogen. The oxygen gets to hog the electrons in those bonds, and the carbon and hydrogen are generally pretty happy with that arrangement. Organisms use a complicated series of steps that allows them to capture the energy from this reaction in the form of ATP, which is then used to power all kinds of cellular processes.

Some of the other comments have mentioned that there are lots of organisms that do not use oxygen. These organisms use redox reactions to get energy, but they use sulfur (or actually sulfate), manganese, iron or other things in place of oxygen. When oxygen is present, organisms that use oxygen tend to dominate, because they can get way more energy than organisms that do not use oxygen.

Photosynthesis is also a redox reaction, but it runs in the opposite direction as respiration. Plants use energy from the sun to power reactions that turn water and carbon dioxide into sugar and oxygen (O2).