What is the name of this part of drawers?
According to this diagram¹ it is a 'mid rail'.

That's a pretty standard North American term. It's quite likely you would find alternates from the UK, Europe, etc. I once worked alongside an architect who was educated in England. This fellow tended to call any horizontal support a lintel although I believe the true definition is a bit stricter.

¹ This diagram seems have at least one typo. 'Gusset or Coner Block' should most likely be 'Gusset or Corner Block'.