It sounds as though they grew from seeds. If they were offshoots, they would have been attached to the rhizome of a parent plant. Yes, they do this in the wild and yes, they will make flowers, but not necessarily like the parent's flowers.

Thanks. Do you happen to know if the height or foliage will be affected or will it remain the same as the parent plant?

A seedling's genes are inherited from the parent(s), so you would expect to see at least some similarities. You might even find that the seedling exhibits characteristics that are indistinguishable from the parent(s), but it is more likely to find variations in seedlings, which is why cultivars are perpetuated through asexual reproduction.

Even if a plant is self-pollenated, its seedlings may exhibit inherited traits that the parent plant has in its genetic makeup, but does not exhibit, or vice versa. There can also be genetic mutations in the seedlings.

Variations in seedlings are most pronounced among cultivated varieties, which result from generations of crosses or selective breeding. In general, the variations in seedlings of cultivated plants tend to favor ancestral species characteristics rather than cultivated characteristics.

The bottom-line answer to your question is "maybe, maybe not."