'Garden-variety' is a horticultural term used to describe plants that are
regarded as native or common, i.e. not exotic or rare. They are the varieties
that you may find in your garden. It is therefore a term that contains a lot
of (British, American, etc.) cultural baggage, because where one's garden is,
indeed whether one has such a thing as a garden, determines which plants are
regarded as potentially being common. The phrase 'common or garden' is
used in the same way. Outside horticultural contexts, the two terms are used
as adjectives for specimens that are plain, unremarkable, standard, etc.
Occasionally 'common or garden' can have a pejorative flavour.
When it is miswritten as 'commoner garden', as it often is, it is probably
because the pejorative sense of 'commoner' (peasant, prole) has seeped into
the phrase.
Another horticultural term that occurs in everyday speech and contains
more cultural preconceptions than is usually noticed is the noun and verb
'weed' (any self-planting plant may be a weed, since it is merely a plant that
you do not want to grow where it is growing).