المقبض على المزهرية.

Because this is a nominal sentence in Arabic. In the present tense, Arabic usually leaves out the verb to be.

So:

• المقبض على المزهرية = The handle is on the vase

There is no separate word for is here.

If you wanted other tenses, then Arabic would use a verb, for example:

• كان المقبض على المزهرية = The handle was on the vase
• سيكون المقبض على المزهرية = The handle will be on the vase

It is a basic nominal sentence made of:

• المقبض = the subject of the sentence (مبتدأ)
• على المزهرية = the predicate (خبر), specifically a prepositional phrase

So the sentence is literally structured like:

• the handle
  • on the vase

This is a very common Arabic pattern.

على is a preposition meaning on or upon.

So:

• على المزهرية = on the vase

As a whole, this prepositional phrase tells you where the handle is. In grammar terms, it functions as the predicate of the sentence.

So although المزهرية is just a noun, the whole phrase على المزهرية works like is on the vase in English.

In Arabic, a noun that comes after a preposition must be in the genitive case (مجرور).

Since على is a preposition, the noun after it is genitive:

• على المزهريةِ

With full endings, the final vowel would be -i.

So if fully vocalized, that part is:

• عَلَى الْمِزْهَرِيَّةِ

In normal unvocalized writing, you usually do not see that ending, but it is still there grammatically.

Because الـ makes a noun definite, like the in English.

So:

• المقبض = the handle
• المزهرية = the vase

This sentence is talking about a specific handle and a specific vase.

Also, there is no problem with having الـ on both words here, because المزهرية is not standing alone as a second noun equated with the first one. It is part of the prepositional phrase على المزهرية.

A fully vocalized version would be:

• الْمِقْبَضُ عَلَى الْمِزْهَرِيَّةِ

More specifically:

• الْمِقْبَضُ = nominative, because it is the subject
• الْمِزْهَرِيَّةِ = genitive, because it comes after على

In careful grammatical reading, those endings matter. In everyday reading and in pause, they are often not pronounced fully.

It is usually pronounced al-miqbaḍ in pause.

A few useful pronunciation points:

• The الـ in المقبض is pronounced al-, because م is a moon letter, so the l sound stays.
• ق is not like a regular English k. It is a deeper sound, often transliterated q.
• ض is an emphatic consonant, often described as a heavy or dark d sound.

In the rest of the sentence:

• على begins with ع, the ʿayn sound
• The final ة in المزهرية is pronounced like -a / -ah in pause, but in full case pronunciation after a preposition it becomes -ti / -ati / -i depending on how fully you are reading the form

That is a very important difference.

• المقبض على المزهرية = The handle is on the vase
• مقبض المزهرية = the vase's handle / the handle of the vase

So:

• على المزهرية expresses location
• مقبض المزهرية is an iḍāfa construction expressing possession or association

The second phrase does not mean the same thing as the sentence you were given.

Yes, but the normal, neutral order here is:

• المقبض على المزهرية

That starts with the subject and then gives its location.

You may also see other patterns in Arabic, but they change the focus:

• على المزهرية مقبضٌ = There is a handle on the vase
• على المزهرية المقبضُ = possible in a marked or literary context, but less neutral

So for a learner, المقبض على المزهرية is the most straightforward order to understand and use.