I got a hosta on sale last fall that is supposed to be T-Rex, but looking how small it is with a tiny leaves I have some doubts about its ID. Generally, is it possible for hosta plant to start as some tiny thing and become a monster in 3-4 years or I just got myself a mini hosta and should review the placement of the specimen ?

i think it's possible but i'm not an expert--my Sum & Substance started with leaves about 5" long from the garden centre, and now 2 seasons later the leaves are about 12"-14". most varieties i have start small and each progressive season the leaves get bigger until peak maturity, which is a little different for each.

my Rainbow's End leaves are about half the size of Blue Mouse Ears right now. it's a baby-baby.

Just to show how tiny it is.

It could well be correct. That is surely juvenile foliage.