Why is it "dog eat dog"?

Solution 1:

Dog eat dog is not a sentence; it's an adjective (and it's usually hyphenated).

You can modify a noun with it:

It's a dog-eat-dog world.

You can also use it as a predicate adjective:

The music business is dog-eat-dog.

Does that still sound ungrammatical to you?

If you are still wondering about agreement within this idiomatic multi-word adjective, perhaps this entry from the OED -- will soothe your mind:

dog, n.1 PHRASES P1. Proverbs and proverbial sayings.
c. [after Latin canis caninam non est ( Varro De Lingua Latina vii. 32)] dog does not eat dog and variants: people of the same calling, origin, etc., do not deliberately harm one another; conversely (let) dog eat dog (cf. dog-eat-dog n. and adj. at Compounds 3a).

There the third-person imperative let accounts for the bare infinitive eat. Here are a couple of examples offered at this entry:

_{1835 W. G. SIMMS Partisan I. v. 59: He cannot escape Travis..who knows the swamp as well as himself. They're both from Goose Creek, and so let dog eat dog.}

_{917 G. L. MORRILL Devil in Mexico 328: Do nothing, let dog eat dog—this is the policy of non-interference.}

Here's another example. An elliptical will accounts for the bare infinitive eat here:

_{1789 Times 19 June 3/1: As it is an established fact, that sharper will not rob sharper, nor dog eat dog.}

_{Definition and examples source: Oxford English Dictionary (login required)}

Did that bare infinitive eat inform the adjectival version? I don't know for sure, but if you still have doubts, then the "It's an idiom!" excuse will have to do.