У меня заканчивается молоко.

Because Russian does not say this idea the same way English does.

English says I am running out of milk, with I as the subject.

Russian structures it more like:

At my place / in my possession, milk is running out.

So:

• у = by / at
• меня = me in the genitive case

Together, у меня often means I have, with me, or in my situation / possession.

So У меня заканчивается молоко is a very natural Russian way to express that your supply of milk is running low.

Because the preposition у requires the genitive case.

The pronoun я changes like this:

• я = I
• меня = me / of me (genitive or accusative in some uses)
• мне = to me
• мной = by/with me

After у, you need the genitive:

• у меня
• у тебя
• у него
• у неё
• у нас

So у меня is fixed and very common.

Because молоко is the grammatical subject of заканчивается.

The verb agrees with молоко, not with я. In other words, Russian treats milk as the thing that is ending / running out.

So the structure is roughly:

• У меня = in my possession / as for me
• заканчивается = is coming to an end / is running out
• молоко = milk

Since молоко is the subject, it stays in the nominative.

Заканчивается comes from заканчиваться / закончиться, meaning to come to an end, to be nearing its end, or in this context to run out.

In this sentence, it means the milk supply is getting used up.

A very literal sense would be:

The milk is finishing / coming to an end.

That is why it works so naturally for things like:

• У меня заканчивается кофе.
• У нас заканчиваются деньги.
• У него заканчивается время.

The -ся makes the verb reflexive, but in many cases reflexive verbs in Russian are not truly reflexive in the English sense.

Here, заканчиваться is the natural intransitive verb meaning something like:

• to be ending
• to come to an end
• to run out

Compare:

• Я заканчиваю книгу. = I am finishing the book.
  Here the verb is transitive: someone finishes something.

• Книга заканчивается. = The book is ending.
  Here the verb is intransitive/reflexive: the book itself is coming to an end.

So in У меня заканчивается молоко, the milk is what is running out, so the reflexive form is the right one.

Yes, заканчивается is present tense.

But in Russian, present tense often describes a process that is currently happening, including one that is approaching completion. So it can mean:

• is running out
• is getting low
• is nearing the end

It does not usually mean that there is already none left. It suggests the process is underway.

If you wanted to say it has already run out completely, you would usually use a past/perfective idea, for example:

• У меня закончилось молоко. = I’ve run out of milk.

This is a very important distinction.

• У меня заканчивается молоко
  = the milk is running out / getting low
  The process is happening now.

• У меня закончилось молоко
  = the milk has run out / is gone
  The process is complete.

So:

• заканчивается = not necessarily empty yet
• закончилось = already finished / no more left

Yes. That is also natural.

кончается and заканчивается are very close here, and both can mean is running out.

However:

• заканчивается often sounds a bit more neutral or standard
• кончается is common in speech, but in some contexts кончать / кончаться can have slang or awkward associations, so many learners are told to prefer заканчиваться in general until they are comfortable with usage

For this sentence, both are understandable and normal:

• У меня заканчивается молоко.
• У меня кончается молоко.

Yes, Russian word order is flexible.

The neutral order here is:

У меня заканчивается молоко.

This sounds natural and unmarked.

But you can change the order for emphasis:

• Молоко у меня заканчивается.
  Emphasizes milk.

• Заканчивается у меня молоко.
  More marked, often emotional or stylistic.

Even when the order changes, the cases still show the relationships, so the meaning usually remains clear.

For learners, the best default is the original order: У меня заканчивается молоко.

It means your supply of milk.

Russian uses молоко without an article, so context does the work that the or some would do in English.

Because of у меня, the sentence clearly refers to milk that you have available for use. So the meaning is not milk as a substance in general, but the milk I have.

That is why English often translates it as:

• I’m running out of milk
• My milk is running out
• I’m getting low on milk

Yes, very often. This is a highly useful pattern.

Examples:

• У меня заканчивается кофе.
  = I’m running out of coffee.

• У нас заканчивается хлеб.
  = We’re running out of bread.

• У него заканчиваются деньги.
  = He’s running out of money.

• У неё заканчивается время.
  = She’s running out of time.

Notice that the verb agrees with the noun:

• заканчивается for singular nouns like молоко, кофе, хлеб, время
• заканчиваются for plural nouns

So this sentence gives you a very useful model: У + person (genitive) + заканчиваться + thing.