The noun 산 is mountain and you can find it in the dictionary. The living/alive must be some transformation from the verb 살다, “to live”.
Many words can share a meaning it is like in english their, there, they're. The only difference is the spelling is the same so it depends on the context
If you say that The translation is alive, then alive should be correct too
That's when you put the noun which is supported by the adjective. I know, in English, writing an adjective without a noun can be grammatical. However, it does not work well in Korean.
I typed out mountain and my keyboard switched it to the emoji ⛰️ and I didn't notice until it said it was wrong.
Now I kind of want to write a list of the emojis with their translations in each language I'm learning haha. I feel like that would be a cute idea if you want to study new vocabulary. Knowing the name would then allow you to search for the emoji when typing in that language too!