Yes, in Russian, the name of the nationality is usually different from the name of the language. That's because nationalities are nouns, English often has the same difference: nationality is Pole, but the language is Polish, Spaniard and Spanish, Swede and Swedish.
The languages are usually named with adjectives, but that's simply because the word "language" is omitted: [the] English [language] ~ английский [язык].
So, "I'm a Pole" is "Я поляк" (nouns in both languages), and "I study [the] Polish [language]" is correspondingly "Я изучаю польский [язык]" (adjectives in both languages).
Note: the only exception is "Russian" (русский), since the nationality is expressed by a substantivized adjective which is identical with the adjective naming the language: "Я русский" vs. "Я изучаю русский [язык]."
The по- words are a bit different. They are adverbs and are used mostly after verbs dealing with speech, such as to speak, to talk, to think, to write, to read (говорить, разговаривать, думать, писать, читать). After such verbs, the по- adverbs can be substituted with a prepositional noun phrase на + language name in the prepositional case:
Я говорю по-русски. / Я говорю на русском [языке].
Я не умею читать по-китайски. / Я не умею читать на китайском [языке].
But when there are no those speech verbs, ot the verb is "to" be, only the на forms are possible:
Эта книга на французском.
Here is a list of some countries with their nationalities and languages. And remember, since nationalities in Russian are nouns and designate people, they all have the other form for females, and also plural forms:
Country: Nationality (male) - Language - In ... language
Россия: русский - русский [язык] - по-русски / на русском
Англия: англичанин - английский [язык] - по-английски / на английском [языке]
Украина: украинец - украинский [язык] - по-украински / на украинском [языке]
Италия: итальянец - итальянский [язык] - по-итальянски / на итальянском [языке]
Франция: француз - французский [язык] - по-французски / на французском [языке]
Польша: поляк - польский [язык] - по-польски / на польском [языке]
P.S. In Russian, the names of the nationalities and languages are capitalized only if they are the first word in a sentence.