To mean 'rainy/winter/summer seasons' in general do we say:
I love the rainy/winter/summer season, or
I love rainy/winter/summer seasons.
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Summer, Winter, Spring and Fall are proper nouns. As such, you would normally just say "Summer", not "Summer season". And of course since it is a proper noun, you wouldn't use "the".
What you may find confusing is that the season names can also be used as adjectives. For example, when discussing TV, you might refer to the "fall season". But in that case, "fall" is an adjective that describes the type of TV season you are referring to.
"Rainy" is an adjective. As such, you can use it to refer to a season. Since "season" is a noun, you need "the", so you would say "the rainy season".
"Season" can refer to one of the four named seasons ("Spring", "Summer", "Fall", "Winter") or any arbitrary period of days that occurs every year ("The Christmas season", "The rainy season", "The baseball season")
Overall, if you say "I love the rainy season", it will sound right. If you say "I love Summer", it will sound right. If you same "I love the summer season", an English speaker would think "summer season of what?"