Basic English use of like with sensory verbs (look, seem, taste, feel, smell, and sound) in similes (comparisons using "like", "as" or "than"). These uses are completely standard and formal, to wit, when written.
to sound like [something]// He sounds like a clown.
to feel like [something]//I feel like a fool.
to seem like [something]//They seem like nice people.
to taste like [something]//This tastes like dirt.
to smell like [something]//This smell like rosewater
This usage holds true for any English (geography-wise, except for pidgins and creoles, etc.) It has zero to do with whether the English is AmE, BrE, Indian, New Zealand, Canadian etc., etc., etc.
And in this sentence:
"To say we’re living through challenging times sounds like both a cliché and an understatement."
One could not omit the like here and keep the meaning of the sentence. "sounds a cliché" doesn't make sense. Sounds, when not used with like, is usually followed by an adjective, not a noun.
- He sounds tired.
- He sounded the fool when he gave his speech. (not usual, but very grammatical)
It refers to speech, so "sounds like" is correct and accurate. After all, speech can sound like any number of things.
There is a difference in meaning between:
- "He sounds angry". [the anger is definite in the listener's opinion
- "He sounds like he is angry". [the listener is comparing the person speech to what anger "sounds like".