Can I get the weapon of another friendly character (with his permission) during combat after he performed his action? I mean, in my turn. If possible, would it be object interaction?
Probably yes but ask your DM
First and foremost — yes you can:
Your character can do things not covered by the actions in this chapter...
When you describe an action not detailed elsewhere in the rules, the DM tells you whether that action is possible and what kind of roll you need to make, if any
PHB page 193, "Actions in Combat"
The only question is — what mechanics the DM will use for adjudicating this? Will (s)he make you to spend an Action or allow you to make it for free? The Player's Handbook implies both options:
You can also interact with one object or feature of the environment for free...
The DM might require you to use an action for any of these activities
PHB page 190, "Other Activity on Your Turn"
We don't know your DM and ultimately it's up to them, but here's my advice.
So your ally ends their turn and gives you permission to get his weapon. Let's simplify the situation a little and assume (s)he drops the weapon at the end of the turn. You can pick up this weapon on your turn without expending an action using a "free object interaction" the rules gives to you. So the mechanics explicitly allows passing a weapon, providing the weapon are not being used between turns. It's up to the players how to describe this action in terms of the narrative.
Now let's say the ally still allows you to get their weapon, did not drop the weapon for some reason. (S)he might want to do this because of two reasons:
- The player doesn't want an enemy to pick up the dropped weapon
- The player wants some kind of mechanical advantage from holding a weapon — a bonus from magic item, a possibility to make an opportunity attack, etc.
Both make sense only when an enemy's turn is in-between players' turns. In this case DM might say "hold on, you can't give a weapon you're actively using right now" and asks the player to spend an action.
PHB describes a precedent of handing an item
Player's Handbook page 190 has a few examples of what "free object interaction" might be:
Here are a few examples of the sorts of thing you can do in tandem with your movement and action:
- hand an item to another character
So the rules assume you can give your weapon to another character without spending an action. They don't describe getting a weapon from another character tho (this still does not mean you can't do this).
D&D 5th edition empowers the DM in ways that 3rd, 3.5, and 4th did not. While rule zero has always applied, 5th edition chooses not to explicitly codify many things. If your DM says you can't, you can't (but don't hesitate to ask "why").