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.
There are generally taxes (e.g., car or gas taxes) that cover the costs. I don't know about Germany's situation, but it is not impossible to make that net zero.
And these comparisons often exclude externalities like carbon emissions, pollution from tires and brakes, road accident costs, noise pollution and the cost of suburban sprawl.
If you would even attempt it to make it net zero you would get truly ungodly amounts of pushback, I don't think you understand how bad it would be. Truly, for those used to a privilege, fairness is an attack.
Furthermore: you have a state of affairs where you are pretty much forced to own and drive a car to e.g. get to work or move around. You make it reflect its true cost making it impossible to afford for anyone except the rich. What now?
