Selling generic products

Hi, I have some generic fitness trackers that another seller has for sale on Amazon under their “brand” although I’m sure they are generic. Should I get my own ASIN numbers and list with those or use the ASIN numbers of the generic that is already listed? My product comes in an unbranded box. I tried listing with the existing ASIN of the supposedly “branded” one but my listing stated it would be dispatched by that brand. This was clearly a mistake and I’ve removed my listing now.
All advice gratefully received.
Chris