Many common plants e.g. tomato, pepper; produce both male DNA carriers and female embrions. In the case that the plant is confined alone, do the resulting seeds carry a carbon copy of the DNA of the parent like when rooting a broken branch in water or do they experience some profound mutations beyond the usual UV induced ones?
A child, in that it's produced by fusing the parent's gametes.
Note that many agricultural and decorative strains derive from "true breeding" or purebred plants - terms that predate a modern understanding of genetics but indicate that the plants are likely homozygous at all the alleles relevant to the observed/desired traits. When you self-fertilize a homozygous plant, the genotype will be the same as if it were cloned. You will still have the plant going through meiosis and therefore recombination processes will still be acting, but there is no variation to yield novel combinations, so in principle it is not much different from a clone (assuming that everything is indeed homozygous).
See also Growing peaches from (supermarket) bought seeds - in many cases, there are actually separate true breeding strains that are then crossed (rather than selfed) to produce the strains you're familiar with. In that case, even in self-fertilizing plants, you can't expect to get similar traits out of the selfed offspring. I believe most commonly used garden tomatoes and peppers are hybridized, so you typically won't want to plant the seeds of the next generation if you're looking for optimal production. Heirloom tomatoes are an exception that tend to be closer to true-breeding as they've been self-crossed for generations.
Some quotes from: https://open.lib.umn.edu/horticulture/chapter/14-4-plant-breeding/
Inbreeding is producing seed by selfing over 5–7 generations to develop plants that are highly homozygous at most loci in their genomes. Homozygosity results in identical plants propagated by seed. This is done so the resulting progeny look more and more alike until, after 4–6 years, plants grown from seeds harvested off the same plant are indistinguishable from each other.
Most new commercial tomatoes, including new garden tomatoes, are F1 hybrids. The seeds you plant in the field are the result of crossing two parents, as described above. (...) Garden catalogs will tell you whether the seed you are buying is hybrid. If you are getting your fruits from the store, you can count on them being hybrids unless they are marked as heirloom.