Some points to note: You said polymers are saturated. This generalization is not true in many cases, e.g. polystyrene, nylon, and much more. Alkanes are always saturated.
Alkanes can have branchings. The ones you have in mind are called normal alkanes, in which no carbon has higher order than two. An alkane can have any kind of branching, as long as it is saturated.
Now, maybe the most important one: polymers are made of so-called monomers. All polymers can be constructed from exactly one monomer (even co-polymers). An alkane which is sort-of tree-like randomly branched, will not have such an unit.
A normal alkane has some similarities with polymers, tho. As others mentioned it, the size is the deciding factor there. A normal alkane with a carbon count of 10000 is for sure a polymer.