Hi, @Chris. Bank accounts themselves can’t be assigned a Class in QuickBooks Online (QBO).
Classes attach to transaction lines, not to balance sheet accounts. Hence, the Balance Sheet by Class shows the net effect of all transactions that carry that Class across all accounts, including bank accounts.
If you filter by a specific Class, you're viewing how that Class has affected all accounts, including bank accounts, through the transaction lines rather than displaying a bank balance broken out by Class.
To see which bank transactions have a given Class, use the Transaction Detail by Account report for that bank account and filter by Class. See screenshot below for visual reference.

For performance by Class on income/expense items, use Profit and Loss by Class.
In short, Classes serve as categories to organize transactions by department, project, location, client, and more. This allows for analysis of profitability and trends without the need to open separate bank accounts.
If you need further assistance or if you have any other QuickBooks concerns, you know where to find me.