Two-Weapon Fighting: When you take the Attack action and attack with a light melee weapon that you're holding in one hand, you can use
a bonus action to attack with a different light melee weapon that
you're holding in the other hand. You don't add your ability modifier
to the damage of the bonus attack, unless that modifier is negative.
In D&D 5e there is no Two Weapon Fighting Style in the way that many other RPG define fighting styles.
Instead you are eligible to make another attack with a off-hand weapon if you met the requirements. In this case you have a light melee weapon in primary hand and a light melee weapon in the off-hand. This makes the character eligible to use their bonus action to make another attack with the condition that your ability modifiers doesn't apply to damage unless it is negative.
If the character is grasping two weapons but doesn't meet the criteria above then he can not take a bonus action to make a off-hand weapon attack per the above rule. Otherwise holding two weapons doesn't have any other effect.
As for the specific tactic the Rogue character use. You have to keep in mind that D&D 5e does not detail which hand is dominant for a PC. Many RPGs, for example GURPS, note that character will have a dominant hand and penalizes attack made with a weapon in the off hand. D&D 5e doesn't deal with this. Hence if you are grasping two weapons, you can do anything that you are eligible to do with either weapon. Including as stated in your example, attack with a Rapier in round one and with the staff in round two. In both rounds the character takes only the action he is eligible for except he using a different weapon held in a different hand.
I will note that because both hands are occupied then both weapons are considered gripped by only one hand. This may be important for specific weapons.