Short Answer: Yes
It was an equipment item in previous editions of D&D, that somehow didn't make it into the current edition. Here is the text from 3.5:
This red silken cord is strong and springy. With a tug, the connected longsword flies into your hand.
A weapon cord chains your weapon to your wrist, so if you drop it (or are disarmed) you merely need to tug on the weapon to bring it back to your hands. You can pick up a dropped weapon connected by a weapon cord as a swift action.
The cord is a little cumbersome, on account of having a weapon dangling from the end of it. When attacking with other weapons while a weapon hangs by the cord you take a -1 penalty for light weapons, -2 for one-handed weapons, and -4 for two-handed weapons. Other items can be attached (such as wands) as a weapon of equivalent size. Attaching or removing a weapon cord is a standard action.
If you have Two-Weapon Fighting you can pull but both weapons as part of the same swift action.
The weapon cord has 0 hardness and 10 hp.
Market Price: 300 gp
Weight: — lb.
This information could easily be converted, but there is no official mention in 5e sources. The penalties here could be left intact, or a general disadvantage can be applied.
If you're looking for the official source, it can be found in the equipment section of the 3e and 3.5e PHB. There, it's an attachment to your armor or gauntlets.
To answer your question at the end, a weapon on a cord or chain probably won't do too much damage to a mount or the character itself. A good example of this was a medieval technique for swords called half-swording, where the soldier would grab the blade of their own sword with their gloved hands in order to more precisely pierce the opponent through armor gaps.
Most weapons rely on the force of a swing or thrust to be able to deal any damage. Also, even with simple leather or cloth, any kind of armor will be able to protect against the hanging weapon. Therefore it shouldn't be a rule that a weapon swinging from a cord be able to do any real damage to anyone