Grappling just says you must use at least one free hand to initiate a grapple. But once you have successfully grappled a target, can you then engage in two-weapon fighting against that target? Or do you have to continually keep one hand free to maintain the grapple every round?
Letting go ends the grapple
Aha! you might say. The rules say I need a free hand to initiate the grapple, but they don't say I have to use my hand to maintain the grapple. I can just let go and have the hand to attack with!
Except, no. Grappling says (emphasis mine):
If you succeed, you subject the target to the grappled condition. The condition specifies the things that end it, and you can release the target whenever you like (no action required).
By putting 'you can release the target' in the same sentence as 'the condition specifies the things that end it', the clear implication (along with verisimilitude) is that releasing the target ends the grapple. In other words, 'See the grappled condition for the list of things that will force the grapple to end, but you can also choose to end the grapple just by letting go.'
Assuming you have just two hands, you will need to release the target in order to have your hand free to employ Two Weapon Fighting. So no, you cannot maintain the grapple while you use Two Weapon Fighting.
Usually, no, as one of your hands must be used to maintain the grapple. Think of it as grabbing their armor/shirt/a part of their body like a monster's spikes. Still, there are certain edge cases where you are able to.
Firstly, if you have additional limbs to use to grapple, you can use them to grapple instead of your two main hands. A Loxodon has a trunk with which they can grapple, same with Simic Hybrids' Grappling Appendages. Thri-Kreen have 4 arms, all 4 of which can wield light weapons. If you wield two weapons in two hands, you still have two hands free, one of which can be used to grapple (If you're planning on doing this, I recommend doing some additional reading on how Thri-Kreen's extra arms work).
Secondly, some monsters are able to grapple with attacks like a bite, thus using the natural weapon they attacked with to grapple, rather than their hands.
Lastly, are abilities which allow you to grapple magically. The first example that comes to mind for me is Coiling Grasp Tattoo, which grapples with magical inky tendrils.