The singular is indicates that you want the reader to think of culture and languages as a single, unified thing. It suggests that the meaning you want to convey by the phrase culture and languages probably doesn't have any clear boundary or distinction between culture and languages. The word something, immediately following, helps reinforce this singular meaning. The fact that language is probably the most central part of culture makes it easy to interpret the phrase this way.
This use of a singular verb to indicate that a subject with plural wording is to be understood as "one" thing is not unusual in English. Your intuition is good. Here are some examples from books:
Love and marriage is the central theme of A Midsummer Night's Dream. [Source]
Afterwards, as Cathy and Brian leave together, a heavy block and tackle falls from the roof of the works section, narrowly missing them. [Source]
If you are someone who wants to have fun making money, while having a healthy and therapeutic job, then arts and crafts is for you. [Source]
Jameson's poignant exposure of a people whose culture and language is suppressed by their invaders stresses an aspect of the tragedy which goes beyond the physical. [Source]
That said, some people will still perceive an incongruity. People's perception of ungrammaticality is not so much based on rules of grammar as on how strongly one sequence of words raises an expectation that another word violates. Block and tackle and arts and crafts occur together frequently, and they describe things that people think of as "one" thing, so it's easy for the pressure of a singular verb to get people to hear them as a singular subject. But since people haven't heard culture and languages together as often, it's a little harder for a singular verb to overcome the expectation that a plural verb is coming. The fact that the plural noun languages comes immediately before the singular verb is adds a little more to the feeling of incongruity.
You could revise the phrase to culture and language. But I see why you want to say languages: the plural "agrees" with—and reinforces in the reader's mind—the fact that you're fluent in several languages. So, to keep languages plural, you can play with another variable: word order. You could rearrange your sentence something like this, which is also more forceful because the stronger, active verb enjoy replaces the fairly inert verb is:
I have mainly studied Economics, but I also greatly enjoy culture and languages. Today I speak fluent Spanish, Italian, and French, and I know how to push subject-verb agreement in English to its limits.
P.S. Here's another question that illustrates how people sometimes get confused about which noun the verb should agree with, based on proximity of words rather than grammar rules.