We know that random sample is a special case of random vector.
Most of the textbooks and online resources defines most of the terminology such as likelihood only on random sample rather than random vector.
Is it only due to the improtance of random sample
or
are there any special terms that can be defined only on random sample but not on random vector?
Just my own thoughts of this.
Lots of things that can be proved for a random sample (i.e. a random vector $(X_1,\dots,X_n)$ such that the $X_i$ are iid) cannot be proved more generally for a random vector.
This is enough already to justify the special attention for samples (comparable with e.g. abelian groups versus groups).
Also random samples are encountered in statistics and are in that sense quite close to what I would call the "real world". Also in "likelihood" I recognize a "real world term".