I should point out first that I don't know the first thing about environmental science, I study mathematics, and the reason I am asking this question is because I want to understand the processes behind climate change better. And, after hearing that methane is such a powerful greenhouse gas, it sounded weird to me that it was also produced during the process of producing biogas, so I basically want to understand how that process works and why the production of methane in it isn't harmful.
When you say produced, do you mean chemically made from another substance, or you do mean released into the atmosphere, or do you mean captured and stored?
Methane, or CH4, is a biogas as it is gas biologically created by organisms. Methane is created in natural processes from which it may be immediately released into the atmosphere or on much slower time scales, such as millennia and more. Methane is also released in some artificial processes, such as the flaring on oil wells. Likewise, in petroleum processing facilities, methane can leak out in other ways than is intended. So, that is how it gets out, or is "produced" as a free gas.
However, if you are asking if CH4 is produced chemically as a product of processing OTHER gases, then the answer is only a mild "sort of". When natural gas from within the Earth is removed from the ground, it contains methane, propane and other gases that are separated into their components. However, most of the nat gas IS methane to begin with. So, saying it is "produced" is sort of true and easily misunderstood.
As Jean-Marie Prival commented, methane is the biogas (after removing impurites such as $H_2S$ and $CO_2$).
Biogas is renewable because "From a carbon perspective, as much carbon dioxide is absorbed from the atmosphere in the growth of the primary bio-resource as is released, when the material is ultimately converted to energy."