on Swiss railways for dogs (not so for cats, birds or whatever) you have to buy half-prized-tickets (the same as for children under the age of sixteen). No I wonder if cats or the dogs should feel discriminated by that rule.
and here a true story combining the themes travel-tickets and animals.
we have a Chalet at the ski-resort Wengen (for 18 years now). access to Wengen is only by train (no cars allowed). local people get special rates to ride the train, also special low-prized winter-season-passes (but only people making their living at Wengen - we don't, nor did our dog Ginger).
We were bored to buy for Ginger at every weekend half-prized-return-tickets and when the next ski-season started we bought (by mail) ski-season-passports for the whole family and asked for Ginger the same (but at the much more favorable "Einheimischen"-tarif = the tarif for the locals).
The administrators of the railways accepted, and for that winter-season Ginger did wear proudly his season-pass with photo and signature (gingers-thumb-stamped with ink!) around his neck - he was THE attraction for visitors and locals and made it even to the local newspapers.