They are not proficient in shields as weapons or armor without feats or multiclassing, and will suffer ASF rolls when casting a spell while using a shield.
Yes, if you multiclass, you would gain proficiency in all armor and shields, and in all martial weapons (shields are martial weapons). You would still suffer any ASF from the shield or armor heavier than light.
A summoner with proficiency can manage a mithral light shield and mithral breastplate without spell failure. The breastplate would provide no benefit while fused, but make a nice backup if anything happens to your eidolon.
Better to just use Extended Mage Armor and the shielded meld abilities, and spend your money on something else like Ring of Protection or Amulet of Natural Armor.