You have exactly 1 bonus action per turn even if you have multiple abilities granting you a bonus action. The exact rule found on the PHB p.189 is:
You can take a bonus action only when a special ability, spell, or other feature
of the game states that you can do something as a bonus action. You otherwise don’t have a bonus action to take.
You can take only one bonus action on your turn, so you must choose which bonus action to use when you have more than one available. (emphasis mine)
To clarify, think of it as having a list of bonus action options on your turn:
- Make an attack with a different weapon (if you meet the requirements for two weapon fighting)
- Second wind
- Some other feature that grants a bonus action
If at any time during your turn you use the attack action and meet the two weapon fighting requirements you are enabling option 1. Even if we consider that using the attack action a second time means adding to the list:
- Make an attack with a different weapon (if you meet the requirements for two weapon fighting)
You must still choose only one of these options.
The wording "a bonus action" is used in many features that grant you a bonus action while not explicitly overriding the rule of allowing only one bonus action per turn. To quote some examples (all emphasis mine):
Rage (PHB p.48)
In battle, you fight with primal ferocity. On your turn, you can enter a rage as a bonus action.
Bardic inspiration (PHB p.53):
You can inspire others through stirring words or music. To do so, you use a bonus action on your turn to choose one creature other than yourself within 60 feet of you who can hear you.
Second Wind (PHB p.72):
You have a limited well of stamina that you can draw on to protect yourself from harm. On your turn, you can use a bonus action to regain hit points equal to 1d 10 + your fighter level.
This is no different than the wording for two-weapon fighting (PHB p.195):
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.
Compare this to Action Surge which states (emphasis mine):
On your turn, you can take one additional action on top of your regular action and a possible bonus action.
This is explicilty overriding the rule that you have only 1 action per turn by using "additional" and "on top of".