Another possible title for this question could be:
Why is integration by parts sometimes called partial integration?
Seemingly something referred to as "partial integration" should be an inverse operation of partial differentiation, but integration by parts is a technique that is relevant even in the most basic single-variable calculus (since it is just a corollary of the product rule and the fundamental theorem).
Moreover, there is already an operation with the name "partial integration" that actually is inverse to partial differentiation. For example, take a smooth function $f(x,y)$, then we can define its partial integrals to be the indefinite integrals $$f_1(x,y)+C_1=\int f(x,y)dx,\quad f_2(x,y)+C_2=\int f(x,y)dy. $$ Then Fubini's theorem can be thought of simply as the integration analog of Schwarz's theorem for mixed partial derivatives. Here is a related question about partial integration.
This answer explains how integration by parts for single-variable functions can be derived by using Fubini's theorem -- does this explain why some people refer to integration by parts as partial integration? Because it can be proved using partial integration?
(As this comment notes that would essentially be the equivalent of calling the product rule "the chain rule" because it can be proved using the higher-dimensional chain rule.)
(For those unfamiliar with this practice -- the German term for "integration by parts" translates word-for-word as "partial integration", and whenever a professor in Germany gives a lecture in English, they almost always say "partial integration" instead of the correct "integration by parts". Although I remember having heard non-Germans make this mistake as well.
For the record, Wikipedia states that the following languages use terms which translate literally as "integration by parts": Bulgarian, Catalan, Czech, Spanish, French, Italian, Macedonian, Polish, Portuguese, Romanian, Russian, Albanian, Slovak, and Ukrainian. In contrast, these languages use terms which translate literally as "partial integration": Bosnian, Danish, German, Indonesian, Dutch, Serbo-Croatian, and Swedish.)
