How about encouraging young people to use, become confident with and
perhaps even enjoy using public transport rather than become accustomed
to always travelling everywhere in an inefficient congestion causing
car? It probably very fair given that all the young people (i.e.
students) concerned don't earn money! Some social justice.

What happened to parents paying their childrens' way, then? Or, in
teenagers' cases, a paper round or Saturday job?

Not everyone in London is middle class. By providing the travel concession
as a universal benefit, children from disadvantaged backgrounds get to
travel free without the stigma of receiving a targeted or means tested
benefit. There are other positive externalities to getting young people used
to using public transport on a regular basis which include the environmental
impact.