How can an organism produced by asexual reproduction be unique and different from its parent?
A mutation in the parent cell's DNA.
In the case of prokaryotes, the parent cell's DNA is copied prior to cell division through binary fission. If there is an error in the replication of the DNA, then one of the daughter cells will be genetically different from the parent cell.
In the case of plants that reproduce through vegetative propagation, the offspring will be structurally unique due to the environment (different soil, nutrients, etc. will affect the number of leaves, etc.).
In the case of fragmentation in animals, environmental factors will also affect certain characteristics. For example, if an earthworm is cut into two unequal halves, the two parts may continue to live as separate organisms. They will be genetically identical, but they may still be structurally unique.