Even here, english has the term 'aping': https://www.google.com/search?q=define%3Aaping, with the same meaning and retains the same animal connotation.
I'm not sure of the origin, but "lust for life" is an expression that's been used in English for at least a century, it seems. It was the name of a biography of Vincent Van Gogh.
The earliest reference on this page seems to be a German film, so maybe it comes from German: https://en.wikipedia.org/wiki/Lust_for_Life
Hmm, maybe it goes back further - Google Books seems to have some instances from the 19th century. A few are negative, basically using the phrase to mean being worldly and not spiritual, a bad thing or even a "curse".
In that sense it wouldn't seem to be the same meaning, but maybe it changed over time.
Then there's plenty examples where a direct English equivalent exists even if it's not a literal translation of the original: nachäffen (de) = na-apen (nl) = mimicking (en), but English lacks the animal connotation (Affe == aap == ape); schwarzfahren (de) = zwartrijden (nl) = fare dodging (en), but English loses the colour in the expression (Schwarz == zwart == black).
Funny thing also that it lists the Spanish "sobremesa" as after-dinner conversation, then lists the Dutch "natafelen" as lingering after having finished dinner, continuing conversation. The two are equivalent expressions, except that one is a noun and one is a verb. Same for "fremdschämen" (de) and "vergüenza ajena" (es), which is known in Dutch as "plaatsvervangende schaamte".