People are good at recognizing particular combinations of sounds
A spell’s verbal component, like any chant or song, is a “particular combination of sounds, with specific pitch and resonance” (PH p. 204).
Just as you can recognize a song even when it’s sung by a different singer, a creature might recognize the particular chant of a spell.
Somatic Components or a spell are also particular
Burning Hands mentions specific somatic gestures in the spell description — casters must touch their thumbs together. There is no allowance for a caster deciding they would rather make another gesture.
I would posit the somatic component of Burning Hands are not unusual, even if it’s unusual that the gesture is defined in the rules. It’s just the example that is called out. Other spells would have their own, particular “forceful gesticulation or…intricate set of gestures.”
Just like Material Components
The material components of given spells are quite particular. A caster cannot just substitute sapphires and a few garnets, instead of rubies. There’s no basis in the rules for assuming the verbal and somatic components are any more free-form. “Particular" (PH, p 2014) is the word.
But leveraging this knowledge is not automatic
Although the components of a spell are particular that does not make them obvious. Often, it is the effect of a spell that is obvious:
“Unless a spell has a perceptible effect, a creature might not know it was targeted by a spell at all. An effect like crackling lightning is obvious...” (PH 204)
Sage Advice backs up the notion that casting a spell is subtle: “Some spells are so subtle that you might not know you were ever under their effects.”
Unfortunately, we don’t really get any guidance on how to determine the how likely it is that the character “might not know” about the casting of a spell — that will be a matter of DM ruling.