Yes, that is what we call an adventitious plant, or daughter plant, and no, your questions are not stupid; we learn by asking.
The true "amazon swords" are in the genus Echinodorus. The commonly-called chain sword and pygmy chain sword used to be classified in the same genus, but these have been moved to another genus, Helanthium. I mention this because the method of reproduction is a bit different vegetatively, and you mentioned runners which is what vegetative reproduction includes. Helanthium species reproduce sexually by flowers and seeds (produced from an inflorescence) when grown emersed, but they also send out runners with adventitious plants. When grown permanently sumbersed they only reproduce via runners in the substrate (or sometimes above it) with adventitious plants sprouting along the way. The Echinodorus species also use sexual reproduction (flowers and seeds) if grown emersed, on what are termed inflorescences. When grown submersed permanently, the inflorescence is still grown but instead of flowers, adventitious plants arise from the nodes along the inflorescence. Rarely do flowers develop on submersed plants, but it can occur; I had it once with E. major which is indeed rare. Echinodorus species never produce runners like the Helanthium species.
You can leave the adventitious plant to develop as is; once it develops white roots and leaves, it no longer obtains nutrients from the parent plant via the inflorescence, but directly from the water column. Sometimes they can be left to grow like this for months; with large parent plants and if the inflorescence reaches the surface, these plants at various levels in the water can be a nice feature. Those closer to the surface tend to develop algae though, due to the closeness of the overhead light.
Alternatively, once leaves and roots are present, the plant can be carefully pulled away and then planted as a new plant. I would wait until the roots are about 1.5 inches or longer, and the leaves are 2 inches or larger.
Another option is to separate the inflorescence itself from the crown of the parent plant, and then stick it somewhere with the adventitious plants still attached. I find that after a while, removing some of the plants will result in overall better growth of the remaining.
Byron.
Edit. Should perhaps clarify the emersed/submersed...these plants are all bog or marsh plants, naturally growing emersed (during the "dry" season) and submersed (during the flood season). They can thus be grown in plaudariums, with the roots in the water and leaves in the air, or permanently submersed. They make ideal submersed plants, which is one reason they are so popular in aquaria. B.
