Learn Swedish: Hästen gillar stallet mer än honom. - The horse likes the stable more than him.

The pronoun you choose completely changes the meaning! Because honom is an object pronoun, it means the horse likes the stable more than the horse likes him. If you used the subject pronoun han, it would mean the horse likes the stable more than he likes the stable. Swedish relies on this pronoun case to show exactly what is being compared when words are left out.

In everyday English, we often say 'more than him' even when we mean 'more than he does'. Is Swedish just as flexible?

No, Swedish is strictly logical about this. If you mean 'more than he does', you absolutely must use han (and you will often hear native speakers add a dummy verb to be extra clear: än han gör). If you use honom, a Swede will always interpret it as the horse having negative feelings about the man.

Wait, isn't mer mostly for uncountable nouns? Why is it next to stallet?

In this sentence, mer isn't modifying stallet (which would mean 'a greater quantity of stable'). Instead, it modifies the verb gillar. It tells us the intensity of the liking ('likes more'). Because actions/verbs are uncountable, you use mer, never fler.

Just a quick refresher: why does stallet end in -et instead of -en?