An item’s caster level affects how vulnerable it is to being temporarily suppressed by dispel magic and the like.
And unless the item is duplicating a spell effect (e.g. spell-completion, spell-trigger), that’s it.
It isn’t a minimum requirement on crafting the item, it doesn’t affect the power of the item’s effect, it doesn’t interact with feats. It interacts with dispel magic and similar, and pretty much nothing else.
If an item is going to affect eldritch blast’s caster level, it will say that, and it won’t have any relation to the item’s own caster level. For instance, an orange ioun stone grants a +1 bonus to caster level, including eldritch blast’s, but the ioun stone’s own caster level is 12th, regardless of what the warlock’s caster level was without the stone or is with it.
Finally, Maximize Spell-like Ability basically pits your caster level with a given spell-like ability against that spell’s spell level—and the caster level is the one you need to have more of. So, for instance, that orange ioun stone could make it easier to use Maximize Spell-like Ability, since your caster level needs to be a certain amount higher than the spell level.
In most cases, spell level never changes—it just is what it is. Eldritch blast is a nearly-unique exception where spell level does change, but it doesn’t change based on the damage that eldritch blast deals. A 20th-level warlock’s eldritch blast is still “like” a 1st-level spell.1 The only way the spell level of eldritch blast changes is if you use a blast shape or eldritch essence invocation on it—then it takes on the highest spell level among eldritch blast’s default 1st, the spell level of the blast shape if any, and the spell level of the eldritch essence if any.
All of which is to say that, normally, any regular casting of eldritch blast with a caster level of 6th or higher is eligible for Maximize Spell-like Ability. The only things that can make this no longer true are a blast shape or eldritch essence invocation applied to the eldritch blast. Feats, items, and so on have no effect on eldritch blast’s spell level, and therefore have no effect on its eligibility for Maximize Spell-like Ability.
- Note that originally, as published in Complete Arcane, this was not true: eldritch blast increased in spell level based on the caster’s warlock level. This was changed in errata, literally in order to make feats like Maximize Spell-like Ability functional with eldritch blast. Even in the original version, though, bonuses to eldritch blast’s damage from sources other than warlock levels would still have no effect on its spell level—only warlock levels increased that.