Es posible que la médica te haya recetado ese analgésico porque eres alérgica a otro.

Because es possible que expresses possibility/uncertainty, and in Spanish that normally triggers the subjunctive in the clause after que.

So:

• Es posible que... = It’s possible that...
• after that, Spanish uses the subjunctive:
  • haya recetado, not ha recetado

This is a very common pattern:

• Es posible que venga. = It’s possible that he/she may come.
• Es posible que hayan salido. = It’s possible that they have left.

In your sentence, the speaker is not stating as a fact that the doctor prescribed it; they are presenting it as a possibility.

Haya recetado is the present perfect subjunctive.

It is formed with:

• haya (subjunctive of haber)
• • past participle (recetado)

So:

• haber → haya
• recetar → recetado
• haya recetado = may have prescribed / has prescribed in a subjunctive context

Why this tense here?

Because the prescription happened before now, but the speaker is still talking about it as something possible, not certain.

Compare:

• La médica te ha recetado ese analgésico. = The doctor has prescribed you that painkiller.
  (statement of fact, indicative)

• Es posible que la médica te haya recetado ese analgésico. = It’s possible that the doctor prescribed / has prescribed you that painkiller.
  (possibility, so subjunctive)

Because médica is the feminine form meaning female doctor.

In modern standard Spanish, especially in Spain, it is very common to distinguish:

• el médico = male doctor
• la médica = female doctor

Some nouns referring to professions used to be treated more often as one form for both sexes, but médica is now standard and very common.

So la médica tells you the doctor is a woman.

Because alérgica agrees with the person being addressed, and here the sentence is speaking to a female.

• eres alérgica = you are allergic (said to a woman)
• eres alérgico = you are allergic (said to a man)

Spanish adjectives usually agree in gender and number with the noun or person they describe.

So if the speaker were talking to a man, the sentence would be:

• Es posible que la médica te haya recetado ese analgésico porque eres alérgico a otro.

Te is the indirect object pronoun meaning to you.

In:

• la médica te haya recetado ese analgésico

the idea is:

• the doctor may have prescribed that painkiller to you

With recetar, Spanish commonly uses an indirect object pronoun for the person receiving the prescription:

• El médico me recetó un antibiótico. = The doctor prescribed me an antibiotic.
• La médica te recetó ese analgésico. = The doctor prescribed you that painkiller.

So te tells us who received the medication.

Ese means that and points to something not associated with the speaker as this would be.

Spanish has a three-way demonstrative system:

• este / esta = this
• ese / esa = that
• aquel / aquella = that over there / that one farther away

So:

• ese analgésico = that painkiller

In real use, ese is very often the normal choice for that.

Also note:

• analgésico is masculine singular
• so the demonstrative must match it: ese

Analgésico means painkiller or analgesic.

It is a masculine singular noun, so the words around it agree with it:

• ese analgésico

If it were plural:

• esos analgésicos = those painkillers

And if you used a feminine noun instead, the demonstrative would change too:

• esa medicina = that medicine

The gender here is grammatical; it does not mean the medicine is “male.” It just determines agreement.

Because porque normally introduces the reason/cause, and that reason is being presented as a real fact:

• porque eres alérgica a otro = because you are allergic to another one

Spanish usually uses the indicative after porque when giving a real reason.

So the sentence has two different moods for two different reasons:

• Es posible que... haya recetado... → subjunctive, because it is uncertain/possible
• porque eres alérgica... → indicative, because this is presented as factual

That contrast is very common in Spanish.

Otro here means another one and refers back to another painkiller / another analgesic.

Spanish often leaves out a noun if it is already understood from context.

So:

• eres alérgica a otro
  literally: you are allergic to another
• natural English: you are allergic to another one

What is omitted is something like:

• otro analgésico

Spanish does this very naturally:

• Quiero este, no otro. = I want this one, not another one.
• Prefiero la roja, no la otra. = I prefer the red one, not the other one.

Yes, but recetar is the more everyday verb for medicines.

• recetar = to prescribe
• prescribir = to prescribe

In medical or formal contexts, prescribir is also common, but recetar is often more natural in ordinary conversation.

So:

• La médica te ha recetado ese analgésico.
  sounds very natural in everyday Spanish.

A version with prescribir is possible, but it may sound more formal or technical depending on context.

Not in this exact meaning.

Te haya recetado is the best choice when talking about a possible action that happened before now and is still connected to the present situation.

• Es posible que... te haya recetado... = It’s possible that she has prescribed / prescribed...

Te recetara is the imperfect subjunctive, and it normally appears in different kinds of sentence structures, usually after a past main verb:

• Era posible que la médica te recetara ese analgésico.
  = It was possible that the doctor would prescribe / had prescribed that painkiller.

So in your original sentence, with Es posible in the present and a completed past action, haya recetado is the natural form.

Yes. With a finite verb form like haya recetado, the pronoun normally goes before the auxiliary:

• te haya recetado

That is the standard placement here.

You cannot attach te to recetado because recetado is a past participle, and object pronouns are not attached to past participles in normal Spanish.

So this is correct:

• Es posible que la médica te haya recetado ese analgésico.

Not:

• Es posible que la médica haya recetádote... ✘

With infinitives and gerunds, pronouns can often attach:

• va a recetarte
• está recetándote

But not with a past participle like recetado.

Yes, it is perfectly natural, especially in a medical context.

It sounds like something someone might say when explaining why a certain medicine was given:

• Es posible que la médica te haya recetado ese analgésico porque eres alérgica a otro.

A native speaker might also use slightly simpler alternatives, depending on tone:

• Puede que la médica te haya recetado ese analgésico porque eres alérgica a otro.
• Quizá la médica te haya recetado ese analgésico porque eres alérgica a otro.

All of these are natural. The original sentence is clear, grammatical, and normal Spanish.