Szeretné, ha együtt koccintanánk és becsomagolnánk az ajándékcsomagot.

Because this is the definite conditional form of szeret.

In Szeretné, ha..., Hungarian treats the following ha-clause as something like an implied azt = that / it:

Azt szeretné, ha együtt koccintanánk...
= He/she/you would like it if we...

Since the object is understood as definite, Hungarian uses szeretné, not szeretne.

A useful contrast:

• Szeretne kávét? = Would you like some coffee?
  → no definite object, so szeretne
• Szeretné, ha maradnánk. = Would you like us to stay?
  → implied azt, so szeretné

Grammatically, it is third person singular, so it can mean:

• he/she would like
• or formal you would like (Ön)

Out of context, both are possible. In real-life usage, many learners will meet this most often as a polite/formal you form.

So the sentence could be understood as something like:

• He/She would like it if...
• or Would you like it if... / You would like it if... in formal speech

Because szeretné, ha... is a very common Hungarian pattern for expressing a wish, preference, or polite desired situation.

Even though ha often means if, in this structure it does not feel like a normal condition in English. It is more like:

• would like it if...
• would like us to...
• would prefer that...

So ha here is part of a fixed, natural pattern, not a literal English-style if clause.

Because after szeretné, ha..., Hungarian normally uses the conditional in the subordinate clause.

The ending -nánk / -nénk means:

• we would

So:

• koccintanánk = we would clink glasses / make a toast
• becsomagolnánk = we would wrap / pack

In this sentence, the conditional does not mainly express doubt. It is part of the normal grammar of expressing a wished-for or preferred situation.

It comes from the verb koccintani.

Breakdown:

• koccintani = to clink glasses, to toast
• verb stem: koccint-
• conditional 1st person plural ending: -anánk

So:

koccintanánk = we would clink glasses / we would make a toast

This is a very Hungarian cultural/social verb, often used when people raise glasses together.

Because együtt means together, and that adds emphasis that the action is done jointly.

So:

• koccintanánk already tells you we would toast
• együtt koccintanánk tells you we would toast together

It is not required for grammar, but it makes the idea more explicit and natural.

becsomagolnánk comes from:

• csomagol = to pack / wrap
• be- = a verbal prefix, often giving the sense of in / up / fully

So becsomagol means something like:

• wrap up
• pack up
• put into packaging

The prefix is attached here because nothing is forcing it to move away from the verb.

For example:

• becsomagolnánk = attached
• nem csomagolnánk be = separated after negation

That separation of prefixes is a very common Hungarian pattern.

Because ajándékcsomagot is the direct object of becsomagolnánk, so it takes the accusative ending.

Breakdown:

• ajándékcsomag = gift package / gift pack / hamper
• ajándékcsomagot = accusative form, the gift package

And:

• az = the

So:

az ajándékcsomagot = the gift package as the thing being wrapped/packed.

This is a very good question, because Hungarian often changes the verb form depending on whether the object is definite.

Here, however, the form becsomagolnánk does not change visibly, because in the conditional, 1st person plural, the definite and indefinite forms are the same.

So even though az ajándékcsomagot is definite, becsomagolnánk is still the correct form.

This is one of those places where Hungarian verb conjugation is doing something important, but you cannot see a difference on the surface.

Yes, but that is completely normal.

Hungarian usually omits subject pronouns when the verb ending already shows the person and number.

Since -nánk means we would, Hungarian does not need to say mi.

So:

• (mi) koccintanánk
• (mi) becsomagolnánk

Both mean we would..., but the pronoun is usually left out unless you want emphasis.

Yes, Hungarian can also use a hogy structure, for example:

Azt szeretné, hogy együtt koccintsunk és becsomagoljuk az ajándékcsomagot.

That is also grammatical.

But szeretné, ha... is a very common way to express a wish in a slightly softer, more tentative way.

A rough difference is:

• szeretné, ha... = would like it if...
• szeretné, hogy... = would like that... / would like us to...

Both are possible, but the grammar after them is different:

• after ha: usually conditional
• after hogy: usually the subjunctive/imperative-type form

As written, with a period, it is a statement:

Szeretné, ha együtt koccintanánk és becsomagolnánk az ajándékcsomagot.
= He/She/You would like it if we toasted together and wrapped the gift package.

But in real conversation, a very similar sentence could also appear as a polite question, depending on punctuation and intonation:

Szeretné, ha együtt koccintanánk és becsomagolnánk az ajándékcsomagot?

So context matters.