Definitely a bad graft with the standard cherry suckering. This happens to a lot of ornamental fruit trees. You'll have to cut the white flowered trunk off at the collar (just past where it meets the main trunk). It sounds like both trunks are the same size? That means this problem started as soon as you planted it. A picture would help a lot. Make a CLEAN cut on an ANGLE just past the branch collar. It is probably too thick to use loppers, so purchase a decent pruning saw (Lowes, HD sell these), and make a starter cut at the bottom of the branch, angling up and in toward the tree, then do the rest of the cut from the top of the branch, angling toward the bottom cut (away from the tree). The cut is on an angle to shed water. You must do a cut on the bottom first so that when you cut from the top it doesn't pull down and rip the bark along the main trunk, which can kill the tree. Do this now or wait until next year and do it in early spring before the buds break. Do not wait much longer, because right now the cherry will use it's energy to heal the wound, but if you wait until May or later it will have used most of it's energy in flowering and growing new branches and will not heal quickly at all. The standard cherry will grow about 20 feet high. Depending on the rootstock, it may even bear fruit. Standard cherries are hardy trees, and it will definitely end up killing your weeping graft if you don't get rid of it soon. As for it looking odd, don't worry, in a few years the weeping tree will grow to the other side with proper pruning. Right now the other tree is preventing it from growing that direction. Once it is gone the other side will start to thrive. The sun exposure alone will help a lot. Don't expect too much this year though, since it will be focusing its energy on healing.

Colt produces pink flowers that smell like melons. Don't know if this is being used outside of fruit orchards. Gisela I haven't seen the top of and known it. Sweet cherry although commonly used is terrible beneath Japanese flowering cherries. I think I've seen that some Snow Fountains ('Snofozam') TM are grown from cuttings.

Other(s) have asked about "weeping" cherries that were purchased smaller-than-usual, that didn't look typical. Yours doesn't either, at this point. Before I think it was determined there was raising from seed involved, probably resulting in a an absence of a weeping habit being passed along to the stock being sold. However, if a single-stem tree is to be made out of yours, without waiting to see if it does it on its own at some point later (looks like it is branching from the base, as trees may often due when planted in the open, without competition for light drawing them up vertically) the procedure is the same: insert a firm stake of adequate height close to the base of the tree (try to figure out where you can due it without pushing down any of the larger roots), select one of the branches and tie it securely into a vertical position at several points along the stake. Use tying materials that will not cut into the bark later, ask at a garden center about this (hopefully the same place you find the stake).

My brother/neighbor here in Saratoga County, Gansevoort has one. Actually it is his second, the first didn't make it. He went back to the local nursery, explained and they replaced it free. It was a little expensive, the blooms were beautiful these last weeks! It is in a raised landscape bed with small shrubs and flowers in full sun. His is a minature. Hope this helps.