I am attempting to explicate and explore the entire meaning of 'sobremesa' in relation to the Spanish culture. I've talked to some Spaniards and they told me that 'sobremesa' is often used with lunchtime and not with breakfast and dinner. Is there a particular reason people do not associate it with breakfast and dinner?
Sobremesa is a well-known Spanish custom but it's not exclusive of Spain. It's practised in many countries and cultures. I'm speaking from my experience in Argentina.
Looking for explanations of sobremesa you'll find that it's often associated with leaving some time for digestion before one goes back to work (or to some other physical activity).
Although it's said that breakfast is the most important meal in the day, most people don't actually comply with the maxim "Breakfast like a king, lunch like a prince and dine like a pauper" (in Spanish: desayuna como un rey, almuerza como un príncipe y cena como un mendigo). Breakfast remains small or nonexistent, and people often need to hurry up for work or school right after it, so there's no need and no time to engage in sobremesa.
There can be sobremesa after dinner, especially if you don't have dinner too late and/or friends and family are in attendance, and if you have time to lose because the next day is not a work or school day. This is obviously not the case most days.
I think the association of sobremesa with lunch (and not breakfast or dinner) stems from the above and from the idea that it's not just waiting for the food to be digested, but also a way to delay your obligations while spending quality time with family or friends. After lunch come work or home chores, so this pleasant delay is important enough to merit a name and a place in culture; after dinner you need only wash the dishes and go to sleep (with or without watching TV before that), so nighttime sobremesa is not considered such an event.