The conversation with Admiral Croft follows soon (a couple of weeks) after Anne’s learning of the engagement in a letter from Mary. Since then, we imagine, she has hardly thought of anything else, because if Louisa is engaged to Captain Benwick, then Captain Wentworth is free to bestow his affections elsewhere:
No, it was not regret which made Anne’s heart beat in spite of herself, and brought the colour into her cheeks when she thought of Captain Wentworth unshackled and free. She had some feelings which she was ashamed to investigate. They were too much like joy, senseless joy!
Note the occurrence of “ashamed” in this passage: Anne does not want anyone to guess her feelings. In this context, when Admiral Croft says:
“Well, now you shall hear something that will surprise you. But first of all, you must tell me the name of the young lady I am going to talk about.”
Anne immediately comprehends that he is talking about Louisa Musgrave, because the engagement has been uppermost in her mind for a fortnight. But she is afraid to suggest the name at this point, because she fears that Admiral Croft might guess the reason why she is so quick to understand, and she would be ashamed if Admiral Croft were to suspect her of being in love with Captain Wentworth. But once he says:
“That young lady, you know, that we have all been so concerned for. The Miss Musgrove, that all this has been happening to. Her Christian name: I always forget her Christian name.”
it is safe for Anne to name Louisa, because there is only one Miss Musgrove whom everyone is concerned for, and so coming out with her name at this point does not give anything away about Anne’s private thoughts and feelings.