Privezak za prtljag ima referentni broj.

A simple breakdown is:

• privezak — the subject noun
• za prtljag — a prepositional phrase describing what kind of privezak
• ima — the verb, has
• referentni broj — the direct object

So the structure is basically:

[Subject] + [Verb] + [Object]

with za prtljag attached to privezak.

Here za means for.

So privezak za prtljag literally means a tag/attachment for luggage.

In Serbian, za is a very common preposition and can have several meanings depending on context, but in this sentence it is the straightforward for meaning.

It actually is in the correct case already.

After za meaning for, Serbian uses the accusative case.
The noun prtljag is masculine singular inanimate, and for many masculine inanimate nouns, the accusative singular looks exactly the same as the nominative singular.

So:

• nominative: prtljag
• accusative: prtljag

That is why you do not see a visible change.

It is in the accusative, because it is the direct object of ima.

The verb imati means to have, and what is had goes into the accusative:

• Ima broj. — It has a number.

Here, though, the object is masculine singular inanimate, so again the accusative looks the same as the nominative:

• nominative: referentni broj
• accusative: referentni broj

So it may look like nominative, but its function in the sentence is accusative.

Because adjectives in Serbian must agree with the noun they describe in:

• gender
• number
• case

The noun broj is:

• masculine
• singular
• accusative here

For masculine singular inanimate, the accusative form of the adjective is the same as the nominative form, so you get:

• referentni broj

If the noun were different, the adjective would change too. For example:

• ima novu etiketu — it has a new label

There novu changes because etiketu is feminine accusative singular.

Because Serbian has no articles.

Unlike English, Serbian does not normally use separate words for a/an/the. Whether something is definite or indefinite is usually understood from:

• context
• word order
• emphasis
• sometimes demonstratives like ovaj, taj, onaj

So Privezak za prtljag ima referentni broj can be understood naturally without any article word.

It can be, but context matters.

Privezak literally means something like:

• pendant
• charm
• hanging attachment
• tag-like attachment

In the phrase privezak za prtljag, it works well for a luggage tag.

Depending on context, Serbian might also use:

• etiketa za prtljag
• oznaka za prtljag

Those can also mean luggage tag or luggage label.
So privezak is understandable and natural, but it is not the only possible word.

Ima is the 3rd person singular present tense of imati — to have.

So:

• ja imam — I have
• ti imaš — you have
• on/ona/ono ima — he/she/it has

In this sentence, the subject privezak is singular, so ima is the correct form.

Yes, Serbian word order is more flexible than English, but this sentence uses the most neutral, standard order:

Privezak za prtljag ima referentni broj.

That is a plain subject + verb + object sentence.

You can move things around in Serbian, but doing so usually changes the focus or emphasis, and some reordered versions may sound marked or unnatural without special context.

For a learner, this original order is the safest and most natural choice.

Yes, prtljag is the trickiest word here for an English speaker.

A rough guide:

• privezak ≈ PREE-veh-zak
• prtljag ≈ PRT-lyag
• referentni ≈ reh-feh-RENT-nee
• broj ≈ broy

A few pronunciation notes:

• Serbian r is usually a tapped or rolled r
• lj is a single sound, similar to the lli in million for many English speakers
• j is pronounced like English y

So prtljag contains:

• prt
  • ljag

It may feel awkward at first, but Serbian allows heavy consonant clusters much more freely than English.

Yes. In Serbian, both Latin and Cyrillic scripts are used.

The Cyrillic version is:

Привезак за пртљаг има референтни број.

The meaning and grammar are exactly the same; only the script changes.