ASME Applies as/if specified by the Project General condition specification or the contract basis of design.
Other codes & standards are also applicable (and sometimes superseeding , supplementing each other , where/as required : can be Norsok , Euronorm , Gost in Russia etc…
Each FPSO or Floating Unit is a different piece of engineering , therefore customized to suit at best the project requirements.
Then you get the regulations and certification that is another field that can influence your codes & standards / piping technical choices (Brasilian NR , Australian AMSA , Russian GOST , Lloyds , DNV , ABS , BV )
Wish it adds clarity in the topic.