Unsurprising, given the close relationship between Yiddish and German, and the fact that the English "lox" is literally the Yiddish word.

The consonants are the same, but whether the vowels are the same depends on your accent. In most British accents the vowel in 'lox' is fairly distinct from that in German 'lachs'. (Or I should probably say 'would be', since it's not a word known to most British English speakers.)

But the German "ch" phoneme doesn't even exist in (American) English?

In this chase, "ch" is pronounced as "k", not as one of the two German ch-phonemes that don't exist in English.

It's not pronounced from the throat like the German "ch" but otherwise it's as close as it gets.

There are (at least?) three different ways "ch" can be pronounced in German: Throaty as in "Buch", a slightly altered "sh" sound like in "Bücher", and just as plain "k", like in "Lachs".

I was very surprised to see wiktionaries phonetic transcription and sound sample of "Buch", apparently it is not throaty everywhere.

It's a /k/ in all "-achs", e.g. "Dachs", "wachsen", "Flachs", exactly as in "bochs" the emulator (compare "boxen"), except across boundaries, e.g. "wachsam, wach-sam" (wakeful, at guard), or contractions, e.g. "[Meister seines].Faches/Fachs".