Aunque yo diga que estoy cansada, mi hermana insiste en acompañarme a la biblioteca.

Because aunque can trigger the subjunctive when the speaker presents something as uncertain, hypothetical, or not especially relevant to the main point.

Here, Aunque yo diga que estoy cansada... means something like Even if I say that I’m tired... or Although I may say that I’m tired.... The idea is not just a plain fact; it is something the sister ignores anyway.

So:

• aunque + subjunctive = even if / although (regardless of whether)
• aunque + indicative = although when the information is treated as a known fact

Compare:

• Aunque digo que estoy cansada... = sounds more like Although I say I’m tired...
• Aunque diga que estoy cansada... = Even if I say I’m tired... / Although I may say I’m tired...

In this sentence, the subjunctive fits better.

Yes, Spanish often leaves subject pronouns out because the verb ending already shows who the subject is.

• diga already tells us the subject is yo

So yo is not necessary for grammar. It is there for emphasis or contrast.

Here, Aunque yo diga... can sound like:

• Even if I say...
• Even though I say...

It gives a little extra stress to the speaker’s point of view.

Because estar is used for states or conditions, while ser is used more for identity, essential characteristics, or classification.

Being tired is a temporary condition, so Spanish uses estar:

• Estoy cansada = I am tired

Using ser here would be unnatural.

A useful shortcut:

• ser = what something is
• estar = how something is

Cansada agrees with the speaker’s gender.

If the speaker is female:

• estoy cansada

If the speaker is male:

• estoy cansado

This is because adjectives in Spanish often change form to match the person or thing they describe.

The verb is insistir en + infinitive / noun.

So:

• insiste en acompañarme = she insists on accompanying me
• insiste en eso = she insists on that

In Spanish, many verbs naturally go with a particular preposition, and you simply have to learn the pattern. With insistir, that pattern is usually insistir en.

So not:

• insiste acompañarme

But:

• insiste en acompañarme

Because after en, Spanish uses the infinitive here: acompañar.

When an infinitive has an object pronoun, that pronoun can be attached to the end:

• acompañar + me → acompañarme

So:

• insiste en acompañarme = she insists on accompanying me

This is very common in Spanish.

You could think of it as:

• en acompañar-me

The attached me means me.

Me is the object of acompañar.

• acompañar a alguien = to accompany someone

So:

• acompañarme = to accompany me

In the full sentence, the sister is the one doing the accompanying, and me is the person being accompanied.

Because ir / acompañar a alguien a algún lugar uses a to mark the destination.

Here:

• acompañarme a la biblioteca = to accompany me to the library

That a means to.

So the structure is:

• acompañar + person + a + place

Examples:

• Me acompaña al supermercado
• Te acompaño a casa

Yes, acompañar often takes the personal a before a person:

• Acompaño a mi amiga
• Ella acompaña a su hermano

But in this sentence, the person is represented by the pronoun me, and object pronouns do not use the personal a.

So:

• acompañarme = to accompany me
• not acompañar a me

Aunque can mean either, depending on context and the mood used after it.

Very roughly:

• aunque + indicative often refers to a real or accepted fact
  
  • Aunque estoy cansada... = Although I’m tired...
• aunque + subjunctive often gives a more hypothetical, concessive, or less important idea
  
  • Aunque esté cansada... = Even if I’m tired... / Although I may be tired...

In your sentence:

• Aunque yo diga que estoy cansada...

This has that concessive sense: regardless of the fact that I say it, my sister still insists.

So in English, depending on style, it can feel like either although or even if, but the subjunctive adds that regardless / it makes no difference nuance.

Both patterns are possible in Spanish, but decir que + clause is extremely common and natural.

So:

• digo que estoy cansada = I say that I’m tired

A more compressed version like digo estar cansada is grammatical, but it is usually more formal or literary in tone.

For everyday Spanish, decir que estoy cansada is the more normal choice.

Yes, it is grammatically possible, but the meaning changes slightly.

• Aunque digo que estoy cansada...
  This sounds more like a straightforward fact: Although I say I’m tired...
• Aunque diga que estoy cansada...
  This sounds more like: Even if I say I’m tired... / Although I may say I’m tired...

The actual sentence with diga suggests that what matters is that the sister keeps insisting regardless of the speaker’s complaint.

So diga is the more natural choice for that idea.