From Wikipedia:
Indiscernibles are objects which cannot be distinguished by any property or relation defined by a formula. Usually only first-order formulas are considered.
Is this because second order systems are categorical and thus every formula is either true or false (I am not sure why I am connecting categoricity with discernibility)? Or can we have indiscernibles at any high order logic level, with this indiscernible becoming discernible at the next order?
First, to clear up a misconception: it is not the case that arbitrary higher-order systems are categorical. Some are, but not all. For example, for any reasonable logic (including $n$th-order, generalized quantifiers, infinitary formulas of bounded rank, etc.) and any signature $L$, there is some $\kappa_L$ such that any theory with models of size $>\kappa_L$ has models of arbitrarily large cardinalities. The vague motto is: large structures do not have categorical theories.
On to the main question. We can talk about indiscernibles with respect to any class of formulas, first-order or not. For example, Otto has looked at a version of the Ehrenfeucht-Mostowski theorem for stationary logic (https://doi.org/10.2307/2275187, https://www.jstor.org/stable/2275187); see also http://intramath.uniandes.edu.co/files/Abstract/(334)-autoXCpag.pdf. The reason we usually look at first-order formulas is that first-order logic is vastly better understood, and frequently the most natural logic for the specific application being considered.
And, addressing the question asked in your final sentence: yes, in general if $L_0$ and $L_1$ are distinct logics then there may be a model $M$ with a set $A\subset M$ which is indiscernible with respect to one but not the other.
