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First - Deutsche Bahn is a private company even if it's shares are owned still by the state - that is due to the financial crisis because it was planned to put the company totally on the market. The company makes profit and even buys companies in Great Britain... ;)
Yes, they get payments from the state as they are the ones that are responsible for maintance of the railway tracks (that are used by other companies, too). And they blow a lot of money in big projects like e.g. Stuttgart 21 whereas the train stations in little towns look terrible since the privatisation... - but I do not want to destroy your wrong ideas...
European policy even includes Germany - and even public transport has to be open to competition... that includes Germany and the German railway system...
A week ticket from Frankfurt to Hanau (about 20 kilometres) costs 46,50 EUR. The month ticket costs 158 EUR. A single fair costs 7,30 EUR - return ticket 14,60 EUR. Compare that with your prices to a town 20 kilometres away into Manchester. I had problems trying that on my own as I do not know your region and your train system - but I do not think that it is cheaper.
The regular fares in Germany are not cheap - it sometimes seems cheap when you have the chance to get special prices for early bookings or special weekend or after business-hour-tickets. Or company tickets if there is special agreements between bigger companies and the transport agency - but then everybody of the company or at least 80% have to take that ticket...
That transportation is included in the ticket prices for matches is calculated into the ticket prices. And the calculation includes that maybe 50 per cent of the ticket owners go by car, that some come from the outer regions of the transport system and some come from the city, that there is special tickets like the "Schönes-Wochenend-ticket" at the weekend with which the fans could go long distances with a group of five for about 28 EUR and that a lot of the fans use the public transportation system every day and already have their month tickets. If you calculate the ticket prices like this and take a little off as group discount... Some save good money with the inclusion into the match ticket - some have to pay more than they get...