The rules do not specify if the flint and steel takes 1 or 2 hands so it is the DM's decision
There are no rules beyond what is described in the entry you quoted. So, when there is something unclear in the rules, it falls to the DM to make a determination in how they want to handle it.
In real life, it certainly takes two hands to use traditional flint and steel
As the link you provide indicates, using flint and steel is indeed a two-handed and often involved process. Though it is one with which your PCs are likely very good at performing.
If you wanted to use real life as the model for your ruling then the best option would be to say that it does indeed take two hands to use a flint and steel.
Regarding holding weapons, they probably would not be able to unless they were using the sword as a steel or rigged up something specifically.
Lighting fires is rarely interesting story-wise so don't worry about the mechanics if you don't have to!
For the most part, it is just assumed that adventurers are capable if not very good at lighting fires. So, under normal circumstances, it doesn't seem super interesting to focus on and track the minutia involved in starting campfires and the like.
As a DM you can make your life easier and your story more interesting by skipping some of this stuff.
If you prefer to do this at your table, or if the characters are in a circumstance that does make it interesting, then use my suggestion above.