What gives brown butter its color?
The Maillard reaction is also called the non-enzymatic browning reaction. The reaction occurs when there are reducing sugars and amino acids present. The reaction is controlled by solvent choice, time and temperature considerations.
When browning butter, certainly, the Maillard reaction is occurring. There may also be other reactions happening such as Strecker degradation and caramelization.
As noted, browned butter is made by essentially heating butter until the water boils off at 100C and then the temperature ramps upward. So, what is causing the browning once the browned particulates are strained out? Some are chemicals that are formed from the reaction.
The color comes about due to production of melanoidins. As the Maillard reaction produces a range of chemical compounds, some of the components may have a lot of color on their own, such as furfural and its derivatives.
Professor Chi Tang Ho at Rutgers University has done extension work on Maillard reactions in model systems. Heath &amp; Reineccius in their book Flavor Chemistry and Technology write about the Maillard reaction in more general terms that includes more on color production as part of the process.