Both are (through intermediaries at least in the case of English) coming from the latin prefix "in-" and "bracchia" (arm).
Also compare to modern Italian "abbracciare".
Just as a side note, 'brace' here (brațe in Romanian, braccia in Italian) is simply the word for arms (as in hands, not as in swords), so the word in romance languages sounds essentially like 'enarm' would in English (without the weapon connotation).
A similar transition is happening with Japanese, where English, the dominant world language in the 20th century and second language for many Japanese, is having its vocabulary widely adopted into the Japanese language. For example, the more common word for "ball" in Japanese is now "bōru" (literally, "ball" pronounced with Japanese syllables), while "tama" refers to more traditional items.
French, interestingly, is consciously hostile to borrowing vocabulary from other languages, to the point of having the Académie française in France [0] (and, to a lesser extent, the Office québécois de la langue française in Quebec [1]) act as "the authority" on the language, prohibiting borrowed words and issuing directives on correct usage and new vocabulary.
[0] https://en.wikipedia.org/wiki/Acad%C3%A9mie_fran%C3%A7aise [1] https://en.wikipedia.org/wiki/Office_qu%C3%A9b%C3%A9cois_de_...
When I was in high-school, I was in a camp with some American Peace Corps students in my country. They were learning Romanian since they had been spending some time here and were curious. At one point, one of them asked 'how do you say "hug"?' to which we replied 'îmbrățișare' (uhm-bruh-tzee-shuh-reh, approximately). They were taken aback, and after a pause, jokingly quipped 'don't you have a shorter word? Like... Hug?'.