Quite similar to claiming to not having or being unable to obtain an id.

Not really.

Prior to the widespread adoption of the automobile and the creation of Social Security, many people never had a need to get a government-issued id - much less one with a photo. So if you became an adult prior to, say, the mid- to late-1960's, there's a non-trivial chance that you might be one of the people who never needed to get a government-issued id. And if you're poor and lack access to a birth certificate, there are obstacles to recreating the baseline documentation necessary to get a government-issued id.

None of that is true for a U.S. Congressman. He's earning close to $200K. Unlike an average person in the 1950's, who has no practical need to get a government-issued id if they don't drive, earning that much money absolutely creates a need to have a bank account. And unlike the small percentage of the elderly population who never got an ID back in the day and who are now rather poor, the Congressman absolutely has the means necessary to get a bank account - he has full-time employment and a large paycheck.