This is a bit of fiction. In most jurisdictions in America, an attorney has an obligation of confidentiality to a potential client during an initial consultation. That's true even if the client or attorney decline to formalize the relationship later on. Anyway, that's the ethical rule as applied to the attorney.

The evidentiary privilege also generally applies in situations where a person is consulting with an attorney and reasonably believes the information exchanged in the consultation is confidential. So in the Breaking Bad episode, Saul didn't really need the dollar to magically "seal" the conversation and have it be treated as confidential.

Interestingly, confidentiality (returning to the ethical rule, as distinguished from the evidentiary privilege) *may* be waived by an attorney to prevent reasonably certain loss of life or substantial bodily harm. Note, the attorney *may* choose to waive confidentiality -- but is not required to!