Can monsters with only Primary Natural Weapons do multiple attacks?
Yes. If they use the full attack action
If you get more than one attack per round [...], you must use a full-round action to get your additional attacks.
This is all included in the OFFENSE section in a monster entry in pathfinder. The fields Melee or Ranged or both will include all the attacks a creature can make. In a standard action attack, they may only make one (normally, there are some special abilities that allow more). If the attacks are a comma separated list, or has the word "and", they can make all the attacks listed in a full attack action.
For example, an adult, white dragon has this listed for Melee,
Melee bite +20 (2d6+10/19–20), 2 claws +19 (1d8+7), 2 wings +14 (1d6+3), tail slap +14 (1d8+10)
The bite and claws are both primary, and the bite also has Weapon Focus, so they are made at the full bab and the wings and tail slap are secondary, so they are made at -5. The dragon gains no additional attacks from a high bab, as these are all natural weapons. The entry has the associated modifiers so you don't need to apply anything else. If the dragon attacked as a standard action (or attack of opportunity or other non-full attack action), it would only get one attack and you would normally use the first entry, in this case, it's bite.
If the entry has attacks separated by "or", then they only get what precedes the or or what follows the or, like the babau demon. In this case, only natural weapons are before the or, and after the or is a manufactured weapon plus a natural weapon.
Melee 2 claws +12 (1d6+5), bite +12 (1d6+5) or longspear +12/+7 (1d8+7/×3), bite +7 (1d6+2)
Note that the longspear has two attack bonuses listed, meaning in this case, the manufactured weapon gains multiple attacks from a high bab. The bite is included in both as the creature can combine manufactured and natural weapons, however, note that the bite, while normally primary, is treated as secondary when the creature is combining natural attacks and manufactured weapons. Also note these modifiers are included, so in general, you can just use the modifiers from the entry.
So, does this means that a monster with multiple primary attacks can use them as per the Two-Weapon fighting rules?
No. Just read the entry. Two-Weapon Fighting rules aren't used with Natural Weapons, at least not normally.
Natural Attacks Most creatures possess one or more natural attacks (attacks made without a weapon). These attacks fall into one of two categories, primary and secondary attacks. Primary attacks are made using the creature's full base attack bonus and add the creature's full Strength bonus on damage rolls. Secondary attacks are made using the creature's base attack bonus –5 and add only 1/2 the creature's Strength bonus on damage rolls.
Without using any manufactured weapons, only the rules for natural weapons are used. No Two-Weapon Fighting rules come into play
Or do I treat them as separate weapons that can only attack once per turn unless it has secondary attacks?
No. They use the above rules, meaning, in a full attack, they make as many primary or secondary attacks as they have in the monster listing. In a standard action attack, they just use one, usually the first. You can tell which are primary as they are the full bab.
The only time Two-Weapon Fighting comes into play are when the creature uses both manufactured and natural weapons
Creatures with natural attacks and attacks made with weapons can use both as part of a full attack action (although often a creature must forgo one natural attack for each weapon clutched in that limb, be it a claw, tentacle, or slam). Such creatures attack with their weapons normally but treat all of their natural attacks as secondary attacks during that attack, regardless of the attack's original type.
But even then, if the creature has more than two appendages that wield weapons, there's a third set of rules, multi weapon fighting. Fortunately, this works just like two-weapon fighting, only they get additional "off-hand" attacks if they have 3 or more appendages that wield weapons.
An example is the girallon
Melee mwk throwing axe +16/+11/+6 (1d8+5), 3 mwk throwing axes +16 (1d8+2), bite +11 (1d8+2) or bite +16 (1d8+5), 4 claws +16 (1d6+5 plus rend)
Similar to two-weapon fighting, with the Two-Weapon Fighting feat, the girallon has multi-weapon fighting, so has a primary attack that gains additional attacks in a full attack action from a high bab, and 3 "off-hand" attacks. However, I don't think this entry is correct... at least not for a standard action attack, bab +12 Str +5, MW +1 = 18, so the standard action attack would be 2 higher than what is listed for the axes, though it is correct for a full attack.
I'm not quite sure why Pathfinder dropped this, but in 3.5, monsters have both Attack and Full Attack listings. In this case (multiweapon fighting), the first attack is lower in a full attack.
For the most part, you can just read the listings. However, if you want to advance an existing monster or create a custom monster, you'll need to understand how all these rules work to create your own listing.