Custom loss function
The concept of a loss function comes from decision theory. Although there are some 'classic' loss functions the point is to be subjective, in the sense of being flexible enough to represent any particular problem conctext. So in that sense, yes, loss functions can be customised.
One of the main ways this has been achieved is via Bayesian regression, as the output of Bayesian regression is a probability distribution which can then the input for a loss function. This blog post shows the mechanics in a simple way (texts for Bayesian statistics and decision are often non-accessible), and also gives some examples of somewhat 'custom' loss functions in 'Examples: Part 3'. -> http://www.statsathome.com/2017/10/12/bayesian-decision-theory-made-ridiculously-simple/