A more fitting complaint about cats would be the Pendlerpauschale, which is paid out to people driving to and from work in their car. It's a much larger sum than the €3b spent on the train ticket subsidy.
It used to be different, but these days that one is paid to everybody who commutes, no matter by which mode. And people who commute by public transport are actually slightly privileged, in that they can claim their actual ticket costs if that works out more advantageous, whereas everybody else is limited to the basic km-dependent allowance, even if their car commutes maybe actually costs more.
And while there are arguments to be had about negative incentives posed by it with regards to incentivising people to commute farther than necessary, the original basic principle is quite simple: Just as companies are mostly only taxed on their profits, so workers ought to pay their income tax only on the money made after deducting any work-related expenses.
And Germany hasn't solved the cost-of-housing problem, either, so a certain degree of commuting is unavoidable. In fact, recently the housing minister basically declared her defeat by announcing that her latest plan for dealing with that problem wasn't building more where people want to live and can make a living, it was subsidising people to move back to the countryside.
