Generally speaking, for AML, all that is truly required is the full legal name and birthday - for KYC more details and identification documents are typically required. 

If a prospective portfolio company wants to be thorough about diligence, they may ask if a Fund has any LPs in certain categories. You may want to ask what KYC/AML service they are using, and that will tell you what they really “need” versus that they “want.” 

Fund managers are obligated to maintain confidentiality with respect to the identities of limited partners - and similarly, certain agreements of the Fund itself are also usually confidential (please check the agreements, as they will vary from jurisdiction to jurisdiction). You may wish to research what are the actual legal obligations for the portfolio company's jurisdiction - for example, for a Delaware domiciled fund, there is no such thing as a formal register of shareholders. 

Formation documents (such as those filed with the Delaware Secretary of State), and information required to run KYC on the ultimate beneficial owners of the a fund's entities (such as the ManCo or General Partner), are sometimes requested, and would not be too unusual.