The verb refers back to the proficiency levels, not the languages, so how many proficiency levels do you have? One, or two?
You could refer to your language skills as one thing no matter how many languages you can speak - but as you are making a comparison, it must be between two things. You are not comparing the languages, but your ability to speak them, so you should use 'levels' and 'are':
My proficiency levels of both the English language and the French language are the same.
You could also group the two languages together this way:
My proficiency levels of both the English and French languages are the same.
This has no bearing on the choice of verb "is" or "are" as this refers back to the 'proficiency levels' not the languages.
Your second option is also a comparison as you are saying two things are equal, so you must use the plural languages.
I am equally proficient in both the English and French language(s).