So this line confuses me a little bit
"If the eidolon only has a single natural attack, the attack is made using its full base attack bonus and it adds 1-1/2 times its Strength modifier on damage rolls made with that attack, regardless of the attack’s type."
Does that mean that even if my Eidolon is attacking with a weapon and has a bite that the bite still gets these modifiers?
No. If you use a weapon attack with the bite in the same round, the bite is now a secondary attack that has a -5 to-hit penalty, and apply only 0.5 str mod to damage for being secondary.
Once you pick up Multiattack, the penalty is reduced to -2 to-hit, but the dmg mod still remains 0.5 str mod.
The weapon attack is at full BAB and with corresponding str mod to dmg for being 1h or 2h, before and after multiattack is picked up.
EDIT : of course you can still attack only with the bite on any given round, and get the 1.5 str mod, while holding your weapon and not using it for an attack. Thus you could have that bite against adjacent opponent on your turn, and still have a reach weapon in hands to take an AoO on an incoming foe.
2nd EDIT : I listed the bite attack as an example, because it's the most obvious way to get 1.5 str to mod, but your quote is from Multiattack itself if i am not mistaken, and could refer to a biped swapping its claws (x2) for a slam for 1 evo pt. Such build is really not playing the strenght of an eidolon (raising the number of natural attacks).