When you say Can I persuade you?, you are saying more than Do you agree with me?. The following are additionally implicated:

• It is not obvious that you should accept what I am proposing.
• If you accept what I am proposing, it is because I have succeeded in persuading you.
• If you admit to being persuaded by me, then you and I are people who see eye to eye, because we work by the same logic.

As is hinted by the excerpt given in the question, the point of using the first variant rather than the second is, aside from informing or establishing agreement, to build intimacy between the two conversants. As other answers have suggested, the phrase could be used ironically when the situation evokes the opposite of intimacy.

When you say “Can I persuade you?”, you are saying more than “Do you agree with me?”. The following are additionally implicated:

• It is not obvious that you should accept what I am proposing.
• If you accept what I am proposing, it is because I have succeeded in persuading you.
• If you admit to being persuaded by me, then you and I are people who see eye to eye, because we work by the same logic.

As is hinted by the excerpt given in the question, the point of using the first variant rather than the second is, aside from informing or establishing agreement, to build intimacy between the two conversants. As other answers have suggested, the phrase could be used ironically when the situation evokes the opposite of intimacy.