There is currently nothing like CRS/AEOI for company registers. However, many countries have signed TIEAs (Tax Information Exchange Agreement) with each other which predate CRS and allows a country to ask another country about the ownership of a company. If French authorities wanted to know the names of the directors, shareholders, and beneficial owners of a company in Mauritius, they would have to file a request for information under the TIEA between France and Mauritius, and Mauritius would have to answer within a reasonable amount of time (a few weeks).
TIEA arrangements have fallen out of favour as CRS focuses on banking and tends to get right to the source, whereas company registers often still don't have proper beneficial ownership registers.
For a while, one way by which OECD measured how reputable a jurisdiction was by how many TIEAs it had signed. Most signed TIEAs with high-profile but low-value jurisdictions like Denmark, Finland, Norway, Sweden, and Iceland just to meet their quota.