Can Clerics use the Spell Focus feat by RAW? The Summoner advice Brian linked me to in his answer mentions that Augment Summoning is a good feat for summoning focused characters, but it requires Spell Focus (Conjuration) as a prerequisite, and my DM is interpreting the reference to schools of magic in the feat text as restricting it to arcane casters, which I'm not sure what to make of because many divine spells are tied to schools of magic as well.
Your DM is wrong
Beneath the spell name is a line giving the school of magic (and the subschool, if appropriate) that the spell belongs to.
Almost every spell belongs to one of eight schools of magic. A school of magic is a group of related spells that work in similar ways. A small number of spells (arcane mark, limited wish, permanency, prestidigitation, and wish) are universal, belonging to no school.
The schools just are; they are not a part of the wizard class, do not refer to actual institutions where one learns wizardry, or anything else. They are nothing but categorized groupings of spells.
Thus, spells always have a school. This is not a property of wizard spells, but all spells, cleric spells included. A cleric can take the feat, and if he does, the saving throw DC of his conjuration spells is improved. For example, heal is a Conjuration (Healing) spell, and undead get a save against it when it used to hurt them. Spell Focus (Conjuration) would make this saving throw more difficult for them.
Do note, however, that Spell Focus (Conjuration) is useless for most summoning spells, since those spells do not have saving throws to benefit from. Thus, getting Augment Summoning without Spell Focus (Conjuration) is usually something summoners try to do; clerics can do this with the Dragon Below domain, which grants Augment Summoning as a bonus feat.
Furthermore, summoning – particularly the summon monster spells – tends to be kind of lackluster. It’s not bad, but the 1-round casting time is problematic, and the actual creatures you get tend to be pretty weak. Summon monster is usually more useful for its utility – the higher-level spells can summon creatures who have spell-like abilities, so preparing those spells is kind of like having one slot that can be any of those spell-like abilities. And wizards – conjurers – are just much better at it than you. And druids get to use summon nature’s ally, which gets many creatures a whole spell level earlier than summon monster does. Just some things to keep in mind.
Yes, a cleric can take the feat Spell Focus
As a feat wholly lacking in prerequisites, nearly any creature can take the feat Spell Focus, even those that can't cast spells at all.
(Maybe such a creature needs the feat to qualify for a prestige class or another feat--like the feat Augment Summoning--or is planning to take levels in a class that casts spells later in its career.)
Your DM is confusing specialist wizards variants with the basic concept of spell schools. Spells would have schools even if wizards variants didn't exist. Almost all spells have a school, including spells that happen to be exclusively divine.
Yes, a cleric can take Spell Focus.