The ''two-weapon fighting'' entry on Beyond is there for every character; it just means that all character can use their bonus action to do a second attack with their off-hand weapon (given specific restriction), not that you have the ''two weapons fighting style''. That is entirely another thing.
- Shields are on the armor list, not the weapons, so you could attack with it, but it is as useful as attacking with a bar stool or your halfling ally: its an improvised attack.
- So, given that you use the Dual Wielder feat (because without it you CANT use the ''two weapon fighting'' action because it isnt light) you would attack without your Proficiency and deal 1d4 dmg without any str bonus to dmg (though your DM can say that a shield used as an improvised weapon could do more than a 1d4 due to its size).
- To ''dual wield'' a sword n board, you would need:
1)Dual Wielder feat -> so you could use the shield as an Off-Hand non-light improvised weapon.
2) Tavern Brawler feat -> so you can add your Proficency bonus to the off-hand attack with an improvised weapon.
3) Two-Weapon Fighting Style -> so you can add your STR mod to the off-hand attack with an improvised weapon.
That's hard to pull-off in RAW. In my games, I changed the shields (parrying dagger, duel gauntlet, buckler, shield and pavise) to be low damage weapons with a special quality that give a AC bonus. So my player can actually fight with both their sword and shield without requiring all the thing specified above.