The insurance company accidentally merged the account details, leading to tens of thousands of pounds - meant for an individual’s retirement fund - ending up in the wrong account.
Despite a number of complaints to the Prudential - including a letter from one of the customers - it took three years for the company to rectify its mistake, which was eventually resolved in 2010.
The Information Commissioner’s Office (ICO), which issued the fine, said the incident was a "serious breach" of the Data Protection Act.
The fine should act as a warning to the financial sector more broadly, the organisation's head of enforcement Stephen Eckersley said.
"Organisations must make sure the information they hold on their customers’ files is accurate and kept up to date in order to comply with the Data Protection Act," he said.
"In this case two customer files were consistently confused and the company failed to remedy the situation despite being alerted to the problem on more than one occasion before it was finally resolved.
"This case would be considered farcical were it not for the serious sums of money involved."
Prudential said the circumstances were "unique" because the customers had the same name and date of birth, adding that it has compensated both customers and apologised to them.
"We regret that this incident occurred and was not resolved more quickly," a spokesman said.
The merging of the account details occurred because of mistaken information given by one of the customer's financial advisers, the spokesman added.
The fine marks the first time the ICO has imposed a monetary penalty for a mistake not related to significant amounts of customers' data being lost by an organisation.