Mindkettő tetszik: az egyik váza, a másik gyertyatartó.

Hungarian often expresses to like with tetszik, which literally works more like to please.

So Mindkettő tetszik is literally closer to Both are pleasing or Both please me.

A few important points:

• the things liked are the grammatical subject
• the person who likes them can be added as nekem = to me
• here nekem is simply omitted because it is understood from context

So:

• Mindkettő tetszik. = I like both.
• literally: Both please me.

Because tetszik is the normal choice for saying that you find something appealing, attractive, nice, or to your taste.

Compare:

• tetszik = I like it / it appeals to me
• szeretem = I love it, I’m fond of it, I like it in a deeper or more habitual sense

With objects like decorations, clothes, pictures, furniture, etc., tetszik is very common:

• Tetszik ez a váza. = I like this vase.
• Tetszik ez a ruha. = I like this dress.

If you used szeretem, it would usually sound stronger or more personal.

Because mindkettő is grammatically treated as singular in standard Hungarian, even though its meaning is logically plural.

So Hungarian says:

• Mindkettő tetszik.

not:

• Mindkettő tetszenek.

This is one of those places where English meaning and Hungarian grammar do not line up perfectly. In English, both is plural. In Hungarian, mindkettő normally takes singular agreement.

Mindkettő means both or both of them and stands on its own as a pronoun.

Examples:

• Mindkettő tetszik. = I like both.
• Mindkettő drága. = Both are expensive.

Mindkét is used directly before a noun, more like both + noun:

• mindkét váza = both vases
• mindkét tárgy = both objects

So:

• mindkettő = stands alone
• mindkét + noun = must be followed by a noun

This is just the normal article rule in Hungarian:

• a before a consonant
• az before a vowel

Since egyik begins with a vowel sound, you get:

• az egyik

Since másik begins with a consonant, you get:

• a másik

So the difference is not about meaning here; it is just pronunciation and article choice.

Together they form a pair meaning:

• az egyik = one of the two / one
• a másik = the other

In this sentence, the speaker is talking about two items and identifying them:

• az egyik váza = one is a vase
• a másik gyertyatartó = the other is a candlestick / candle holder

This pair is very common in Hungarian when exactly two things are being contrasted.

Yes, from an English point of view, the equivalent of is is missing.

Hungarian normally leaves out the present-tense third-person copula in simple sentences like this:

• Az egyik váza.
• A másik gyertyatartó.

English must say:

• One is a vase.
• The other is a candlestick.

So this is not incomplete Hungarian. It is normal Hungarian grammar.

Because Hungarian often allows a very compact, label-like style in predicate nouns, especially in short contrasts like this.

So:

• az egyik váza, a másik gyertyatartó

is a neat, concise way to say:

• az egyik egy váza, a másik egy gyertyatartó

Adding egy would make it a bit fuller and more explicit. Leaving it out makes the sentence more compact and elegant.

Hungarian word order is strongly shaped by information structure, not just by subject-verb-object order.

Starting with Mindkettő sets up the topic:

• As for the two of them, I like both.

Then the part after the colon explains what the two things are:

• one is a vase
• the other is a candlestick

So the sentence is organized as:

1. main statement: Mindkettő tetszik
2. explanation/details: az egyik váza, a másik gyertyatartó

That is a very natural Hungarian way to present the information.

Yes, váza here is simply the basic noun vase.

Some Hungarian nouns naturally end in -a or -e in their dictionary form. That does not automatically mean possession.

So:

• váza = vase
• vázája = his/her vase

A learner might suspect that váza means its vase or his/her vase, but it does not. In this sentence it is just the ordinary noun.

Yes, it is the same verb form, but here it has its ordinary lexical meaning: to please / to be liked.

For example:

• Tetszik ez a kép. = I like this picture.

In polite Hungarian, tetszik can also appear in special polite expressions, such as:

• Mit tetszik kérni? = What would you like?

So the form is the same, but the function is different:

• here: normal verb meaning to please
• in polite speech: part of a politeness formula

That can definitely confuse learners at first.