A regressive tax charges small companies a higher PERCENTAGE of income than large companies, which is exactly what a flat fee would do.
Flat tax generally means a flat percentage not a flat fee. Regressive tax is where the rate (percentage) goes down as you scale up. A tax that is a fixed total amount (not a fixed total percentage) is regressive because the same amount is a lower percentage of a higher income. You can still charge the rich total more than the poor and have it be regressive; eg a regressive tax where you charge 10k income at 10% and 1k income at 20% you still charging the rich a larger amount.
Or to give the US example, sales tax (and 'vice' taxes like cigarettes and liquor) - because of the different spending habits of different groups, they're effectively regressive taxes.