The translation might not be technically bad. "Engage" and "engagement" are general terms about joining together (for example, the teeth of the gears on a bicycle engage with the drive chain), and one of the contexts that these words are used is for war - a "military engagement" is another name for a battle (because the opposing forces meet on the battlefield).
However, in this case, it is definitely about a marriage between a man and a woman (so, yes, it was good that you noticed! XD).
Incidentally, "marriage" has a similar meaning as "engagement" and it can also be used in general terms (meaning joining, such as in car manufacturing/assembly, where the engine is said to be married to the chassis/body). However, although this general use is technically correct, it is quite rare in normal conversation, where marriage almost always means a wedding between a man and a woman.