Such a shock for me (a native speaker who has never ever used the form "yaşımdayım") as well!

Yes but "25 yaş" means "25 years" as a quantity.

If you want to point out to a specific year youn eed to say "25nci yaş"  meaning "25th year".

"25 yaşı" means "year of 25".

So I would parse it as:

25 yaşındayım = I am at the year of 25.

And by the way "yıl" and "yaş" are both the same word in Turkish meaning "year". The same word mutated over years in different ways.

There are three different interpretations for the reference of the POSS in this thread now.

1. The years belong to the one who lived them says scalpel.

2. The years are earmarked because they appear in a compound says si++.

3. The years belong to an abstract concept of age say tomac and Abla.

tomac, I have a feeling we are going to lose this.

If you google "yaşındayım yaşımdayım" you get many natives´ discussions about this issue. Too complicated for me. I even found a poll about which alternative is correct.

Yes, that´s the gist of the issue.. when an adjective modifies a noun there is no need for the possessive suffix.. 

oda kapı-sı (noun compound) 

Yeşil kapı (adjective compound)

As the numbers are adjective, they should follow the rule..

2 saat

5 hafta

3 yıl

25 yaş

Let´s use them in sentences..

Deney 2 saat sürdü

5 hafta sonra tekrar karşılaştılar

İnşaat 3 yılda bitti

Bedelli askerlik 25 yaşla sınırlı

They may appear in noun compounds.

Bunun (2 saat) i

Sürecin (ilk 5 hafta) sı

İnşaatın (3 yıl) ı

(25 yaş   ı-n-ı bitirme şartı

As you can easily see, the possessive suffix at the end belongs to the noun that is placed at the beginning of the group. (there is an inlaid pronoun in the last sentence: (onların /25 yaş/ ı.. )

From this point of view, "25 yaşı" is grammatically impossible on its own.. and I am wondering how a native with a good grammar knowledge claims it to be so!. 

On the other hand, it is grammatically correct to say "25 yaştayım" which actually means "I am in the age of 25", but we never use it.

Not using 1st person possessive, may cause some problems. For example, 

Ankara´ya ilk kez 5 yaşındayken gittim is deemed to be clear even if it is not yaşımdayken, but 

Ali´yi ilk kez 5 yaşındayken gördüm is not clear.. Whose age is 5? Ali´s? or the speaker´s? Without the context 51% would claim it is Ali´s, and 49% would claim it is the speaker´s..

If it is the speaker´s age, we need to rewrite sentence with using -ım instead of -in..

Ali´yi ilk kez 5 yaşımdayken gördüm.

If using yaşımdayken solved the problem why shouldn´t I use it? What is grammatically wrong with it? We are free to say evimdeyim, for instance, köyümdeyim, sözümdeyim, odamdayım, but not free to say yaşımdayım.. interesting eh? 

Which is grammatically correct

Ali benim yaşında or Ali benim yaşımda?

Those who claim it should be the former, have to explain how 2nd person poss can follow f1st person poss: benim yaşın!

I wonder if they also say benim evin, benim odan, benim fikrin, benim geleceğin, etc, etc, etc,

..

As I said in my first post to this thread, using yaşın instead of yaşım in some cases is a common error and this error is so common that even a grammar man can commit it and never realize if it is actually a mistake..

yaş+ım+ı (never yaşını with 1st p.sg)

yaşımı sorma, yaşımı tahmin et.. 

yaş+ım+da (never yaşında with 1st p.sg)

yaşımda ne varmış? bu yaşımda bunu da gördüm..

* a sentence like, "bu yaşında Ankara´ya gidemem" is incorrect, it should either be "bu yaşta" (in this age) or "bu yaşımda" (in this age of mine) 

* a sentence like, "benim yaşında birisi" is a crime against the Turkish Language

yaş+ım+ın (never yaşının with 1st p.sg)

yaşımın gereği, yaşımın nesi var?

yaş+ım+dan (never yaşından with 1st p.sg)

yaşımdan utandım, yaşımdan sana ne?

* a sentence like, "bu yaşından sonra Ankara´da yaşayamam" is incorrect, it should either be "bu yaştan" or "bu yaşımdan".

yaş+ım+a (never yaşına with 1st p.sg)

yaşıma bakma, yaşıma göre dinçim..

* a sentence like, "bu yaşına kadar Ankara´da yaşadım" is incorrect, it should either be "bu yaşa" or "bu yaşıma".

...

Last words for the learners.. I just explained which is grammatically correct.. but, of course, you are free to make your choice  

I guess it shold be me. So I should explain.

If it is claimed "25 yaşım" to be grammatically correct then "25 yaşı" should be correct as well right?

Both of them have a possessive suffix in the end.

And nobody can deny the existence of the following verb:

x yaşında olmak = to be x years old

Just apply personal endings to it:

x yaşında-yım = I am x years old

x yaşında-sın = you are x years old

x yaşında = he/she/it is x years old

x yaşında-yız = we are x years old

x yaşında-sınız =you are x years old

x yaşında-lar = they are x years old

I don´t think I have ever said "x yaşımdayım" so far and I don´t think I ever will. My brain just doesn´t work that way.

If it is claimed "25 yaşım" to be grammatically correct then "25 yaşı" should be correct as well right?

Both of them have a possessive suffix in the end.

Good point.. Let´s see where it will lead us to

First, remember,  I said "on its own"..

No adjective compound takes a possessive suffix.. unless modified by a noun..

For example, it is always "yeşil kapı"..

If you say "yeşil kapım", you have to accept that there is this 1st p.poss.adj "benim" (inlaid or not) modifying the group:

benim /yeşil kapı/ m. 

And this is no more an adjective compound but a noun compound..

Likewise, "yeşil kapısı" needs "onun" : onun yeşil kapısı

Good.. now, let me go back to "25 yaş"..

Of course, 25 yaşı is possible too..

But the thing you forget is this 3rd p.poss. adj that you need to use to modify the group: onun..

In other words, "(onun) 25 yaşı" is for 3rd.person singular. 

This is where it leads us to  And you think 3rd p.poss can be applied to 1st person?

Ali 25 yaş-ı-n-da  (n is buffer letter)

Let´s see how it works with 2nd p.sg:

Sen 25 yaş-ın-da-sın (-ın is poss for 2nd person)

So, it is expected that 1st p.sg should take its own suffix: -ım

Ben 25 yaş-ım-da-yım

Q- Why 1st p.sg sould take -ı or -ın but not -ım which is its own?

And nobody can deny the existence of the following verb:

x yaşında olmak = to be x years old

No, there is no verb listed in the dictionaries as "x yaşında olmak" 

Don´t make up a new verb to describe/support your idea

x yaşındasın = you are x years old

x yaşında = he/she/it is x years old

x yaşındasınız =you are x years old

x yaşındalar = they are x years old

What confuses you all is that "yaşında" is used for both 2nd and 3rd person..

But as I explained above.. this resemblance is in appearance..

2nd p.=> yaş-ın-da- 

3rd p. => yaş-ı-n-da 

I don´t think I have ever said "x yaşımdayım" so far and I don´t think I ever will. 

You don´t (and will never) say "yaşımdayım", but you say "yaş-ım" and "yaş-ım-da" ? 

Q- If the answer is yes, what is wrong with adding "-y-ım" to it? (yaş-ım-da-y-ım )

If the answer is no, I will be really looking for your explanation that I´m sure it will be very interesting..

My brain just doesn´t work that way.

I think, it´s nothing to do with brain.. if it was up to the brain/knowledge of grammar, we all would know it´s an error.. but apparently most of us don´t know..

There must be a reasonable explanation for why we often prefer "yaşındayım" to "yaşımdayım"..

But just because it is preferable doesn´t make it grammatically correct.. that´s my point..  

What if the common use of 3^rd person POSS here is a result of ellipsis:

                            (hayatımın/yaşamımın/yaşlarımın) 25 yaşındayım ?

Would it make any sense? I mean only in this particular idiomatic case where age is told. There has to be an explanation why so many natives feel it’s ok even though there is a grammatical problem in it.

...but then it would be: yirmibeşinci =>25th

Hayatımın 25. yaşındayım (hayatımın yirmibeşinci yaşındayım)

I agree.. "there has to be.."

But nobody (from the other camp) has yet come up with a good explanation  

It´s right there! And everybody using it. Just google "yaşında olmak" and you will see many hits.