Hard questions to answer.....I like that.
You are correct, for registration and certification, there are often several choices. Some economies do in fact have multiple accreditation bodies for lab accreditation also. The North American market is the most populated there, with five recognized bodies (IAS, A2LA, NVLAP, L-A-B, SCC/CLAS). Another economy with more than one AB is People's Republic of China (PRC) who technically has two ABs (CNAL, HKAS).
In the cases where the only AB is government owned, then typically the AB is affiliated with the National Measurement Institute (NMI) for the country. In the U.S. for example, NVLAP is affiliated with NIST.
Hope this helps a bit.
Hershal