On the graph, you can clearly see the seasonal demand for power during the summer months as air conditioning kicks in. The electricity production for gas mimics this as well each year. But the solar output does not seem to have a spike each summer. Why is that? You would think that solar output would be significantly higher when the summer months are here.

It's global data, "here" is not US, and it's solar+wind, not only solar. Summer months have a different definition depending on hemisphere and solar production maximums depend heavily on latitude.