From a plumber's perspective: if your shower or bath is slow to drain and other drains are OK (e.g. toilet or kitchen), it appears the block is local to the bathroom. The suspicion of a local block increases when you are in a unit that is on an upper floor.
Sometimes when we go to unblock a slow draining bathroom, we are able to open up the waste grill and pull out a nasty mess of hair/soap scum etc. Once this is removed, the bathroom drains runs freely. In this instance, it pretty much appears that the resident is at fault so perhaps the agent has a good argument.
As you say though, it's a bit hard to say if the hair is yours or not.
Another scenario is that we suspect the block is local to the bathroom but we can't pull it out. We can only push it down and away. Its a bit hard to say then definitively what blocked the pipes.
If these things are happening in a ground floor unit and other drains are affected (e.g. toilet slow to empty) then you more likely have a pipe or foreign object problem that is causing issues in one of the larger common pipes and backing up into ground level units.
I guess it is a bit frustrating for ECs and/or landlords to pay for unblocking bathrooms when they are blocked with hair so if it keeps happening in units you live in, grab a plug filter or strainer for the bath/shower. This should save you and your landlord the grief.
If a local block happens again, you may struggle to get out of paying a second time?