Wielding two weapons without “Two-Weapon Fighting” or “Dual Wielder”?

We have a Barbarian on our team that very much likes to switch up his weapons; he goes from a Dwarven Hammer, to a pair of Hand-Axes, to occasionally just going bare-fisted. No issues there, he’s fairly adept with all 3.

However, I have only just recently picked up on the way that he rolls his attacks for the two Hand-Axes: He rolls the first attack, adds his modifiers etc. Then he rolls the second attack, this time without modifiers, which is all well and good. But I’m not sure if this action uses his bonus action or not, or is considered a separate action (though, being a level 4 Barbarian, he only has one action per turn), or is merely part of the same action.

If we have a look at the Two-Weapon Fighting feature from the Fighter’s fighting styles:

Two-Weapon Fighting
When you engage in two-weapon fighting, you can add your ability modifier to the damage of the second attack. (PHB5e p.72)

Similarly, the Dual Wielder feat also uses the Two-Weapon Fighting feat:

Dual Wielder
You master fighting with two weapons, gaining the following benefits:

  • You can use two-weapon fighting even when the one-handed melee weapons you are wielding aren’t light. (PHB5e p.164, emphasis mine)

So; yes, this gives a bonus to the second weapon attack, but having asked a separate question to clarify which action this secondary weapon attack belongs to, it was determined that this uses your bonus action;

[…] the second attack requires a bonus action

So after that long winded explanation, here is the question: Without any additional feats, can you wield and use two weapons at once, and if so, how many actions does it require to make a full attack with both weapons in a turn?