Can Ramadan or Shawwal be considered a date for paying Zakat in the event of not remembering the exact date the money reached the nisab? And must it be paid on a specific day of the month?

Zakat is a great pillar of Islam, and it must be given care and its rulings known, along with precisely determining the time of reaching the nisab (minimum taxable amount) and the time of its obligation.

Zakat is not obligatory based on doubt; the default is that it is not obligatory. It becomes obligatory only when the nisab is definitively reached. If you are in doubt between Ramadan and Shawwal, then make your zakat hawl (one lunar year) in Shawwal.

It is obligatory to pay zakat immediately on the day it becomes due, provided one is able to do so. It is not permissible to delay it without a valid excuse. This is the view of the majority of scholars.