@Hector M Huerta Jimenez , Thanks for posting a good question.
This looks more of a general certificate handling question, kindly take a look at the Microsoft document Certification Authority Guidance', it is an old doc ' but similar concepts still apply.
It could depend on the scenario and type of clients that're connecting to the server. To be specific, if you are using a cert just for securing a website/app, typically it doesn’t affect if you change the organization name.
You may wish to know "DigiCert will deprecate the Organizational Unit field"
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