My mom and I were talking at dinner this evening, and she mentioned that a friend of hers knew a friend that had to have their throat stretched. (Yes, I know, the whole friend of a friend deal) But apparently, my mom’s friend was convinced and claimed that she even knew of several other people that had to get their throats stretched.
People who’ve had damage to their esophagus (from prolonged acid reflux, or from other causes) that leads to scarring can develop an esophageal stricture which makes swallowing difficult. Treatment involves dilating the stricture, so it doesn’t interfere with swallowing any more. I suspect that may be what your mother is referring to when she’s talking about “throat stretching”.
To go with the factual links I’ll add the anecdotal evidence that my mother has had her throat stretched twice. Both times in consequence of getting a piece of steak caught in her throat, causing damage.