Şirket aynı şeyi iddia ediyor.

Because aynı şeyi is the direct object of the verb, and it is specific/definite: the same thing.

In Turkish, a specific direct object usually takes the accusative ending.

• şey = thing
• şey-i = the thing / that thing, as an object

Because şey ends in a vowel, Turkish inserts a buffer y before the ending:

• şey + i → şeyi

Also, notice that the case ending goes on the noun, not on aynı:

• aynı şey = the same thing
• aynı şeyi = the same thing, as the object

Compare:

• Şirket bir şey iddia ediyor. = The company is claiming something.
• Şirket aynı şeyi iddia ediyor. = The company is claiming the same thing.

İddia ediyor is the present-tense form of the compound verb iddia etmek, which means to claim or to assert.

This is a very common Turkish pattern:

• a noun or verbal noun
• plus etmek
• together functioning like one verb

So:

• iddia = claim, assertion
• iddia etmek = to make a claim / to claim
• iddia ediyor = is claiming / claims

Even though it is written as two words, you should learn iddia etmek as a single vocabulary item.

This is a normal sound change with etmek in many forms.

The verb etmek often becomes ed- before vowel-initial endings:

• etmek
• ediyor
• etti
• edecek

So the correct form is:

• iddia ediyor

not:

• iddia etiyor

This is something you will see again and again with verbs built with etmek.

Because the most neutral Turkish word order is Subject + Object + Verb.

So this sentence breaks down as:

• Şirket = subject
• aynı şeyi = object
• iddia ediyor = verb

That gives:

• Şirket aynı şeyi iddia ediyor.

English usually prefers Subject + Verb + Object, but Turkish usually puts the verb last.

Other word orders are possible in Turkish for emphasis or contrast, but this version is the standard, neutral one.

Turkish does not have articles like English the and a/an.

So şirket can mean:

• the company
• sometimes company in a more general sense, depending on context

If you want to make it clearly indefinite, you can use bir:

• bir şirket = a company

In your sentence, şirket most naturally means the company because the context usually makes it clear which company is being talked about.

It can correspond to either one, depending on context.

The suffix -iyor is often taught as the present continuous, so:

• iddia ediyor = is claiming

But in real Turkish, -iyor is also often used for actions that are current, ongoing, or repeatedly happening in the present, especially in reporting and news-style language. So English may translate it as:

• is claiming
• claims

Both can work.

A more general or habitual form would be:

• iddia eder = claims, would claim, tends to claim

But in many everyday and journalistic contexts, iddia ediyor is very natural.

Because aynı is being used as an adjective before a noun.

Here:

• aynı şey = the same thing

So:

• aynı describes şey

By contrast, aynısı means the same one / the same thing and usually stands on its own, without a following noun.

Compare:

• aynı şey = the same thing
• aynısı = the same one / the same thing

So in your sentence, since şey is present, aynı is the correct form:

• aynı şeyi

Yes, it can, if the context already makes the subject clear.

Turkish often leaves out subjects when they are understood from the situation. So you could say:

• Aynı şeyi iddia ediyor.

That could mean:

• It is claiming the same thing
• He/she is claiming the same thing

The problem is that the 3rd person singular verb form does not tell you exactly who the subject is, so if the context is not clear, Turkish often keeps the noun:

• Şirket aynı şeyi iddia ediyor.

So including şirket makes the sentence clearer.

A useful breakdown is:

• Şirket = company
  
  • the subject
• aynı = same
  
  • an adjective
• şey-i = thing + accusative
  
  • the direct object
• iddia ediyor = is claiming / claims
  
  • the verb phrase

So the structure is:

• Şirket
• aynı şeyi
• iddia ediyor

or, in grammatical terms:

• Subject
• Object
• Verb

This is a very typical Turkish sentence pattern.