Yes people do use this phrase but it is somewhat associated with old fashioned style - especially with regard to face to face selling.
Indeed you're likely to find parodies of old style shop interactions 'Could I interest Sir in one of our fine silk ties...?" If you search Youtube.
I'm pretty sure Monty Python did several.
I would not be surprised if an older person (60 plus) used this phrase with me but someone younger would only use it in jest or parody I think.
The more common use of the verb is in the other format which you correctly distinguish from this
' I am interested in ...'