Mi hermana es igual de paciente que mi profesora.

Because paciente here is being treated as a general quality, and Spanish normally uses ser for that.

• ser = what someone is like, in a more general or defining sense
• estar = how someone is, often in a temporary state or condition

So:

• Mi hermana es igual de paciente que mi profesora = my sister is as patient as my teacher, as a quality
• Mi hermana está igual de paciente que mi profesora would sound unusual unless you were talking about a temporary situation or behaviour at a particular moment

In most normal contexts, es is the right choice.

It is a comparison structure.

The pattern is:

igual de + adjective + que

So here:

• igual de = equally / as
• paciente = patient
• que = than/as, introducing the second person in the comparison

So the sentence structure is:

Mi hermana + es + igual de paciente + que mi profesora

This means that your sister and your teacher have the same level of patience.

Yes. That is very natural Spanish too.

Both of these mean basically the same thing:

• Mi hermana es igual de paciente que mi profesora
• Mi hermana es tan paciente como mi profesora

A useful way to remember them is:

• tan + adjective + como
• igual de + adjective + que

Both are correct. Many learners meet tan...como first, but igual de...que is also common and perfectly normal.

Because Spanish comparison patterns use different words depending on the structure.

For example:

• más... que = more... than
• menos... que = less... than
• igual de... que = as/equally... as
• tan... como = as... as

So you cannot freely swap que and como. You need to learn the full pattern.

That is why this sentence has:

igual de paciente que

not

igual de paciente como

Because in Spanish, a short possessive like mi, tu, su, nuestro, etc. normally replaces the article.

So Spanish says:

• mi profesora
• mi hermana

not:

• la mi profesora
• la mi hermana

English does the same in a way: you say my teacher, not the my teacher.

Because paciente is one of many Spanish adjectives that have the same form for masculine and feminine in the singular.

So you get:

• mi hermano es paciente
• mi hermana es paciente

The adjective stays the same.

Only the plural changes:

• mis hermanos son pacientes
• mis hermanas son pacientes

This is common with adjectives ending in -e and with many ending in a consonant.

No. It is not masculine in the sense of agreeing with hermana.

In this structure, igual de works as a fixed comparison expression, and igual does not change form.

Also, even when igual is used as an adjective, it often has the same singular form for masculine and feminine:

• un resultado igual
• una situación igual

So igual does not become iguala.

Profesora is the feminine form of profesor and means a female teacher or professor.

In Spain, profesor/profesora is very commonly used for school teachers, especially in secondary education, and also for university teachers. In some contexts, especially primary school, you may also hear maestra.

So in a basic sentence like this, mi profesora is best understood as my female teacher.

You can, but it is slightly different in structure and feel.

• Mi hermana es igual de paciente que mi profesora directly compares the degree of patience.
• Mi hermana es paciente, igual que mi profesora means something more like My sister is patient, just like my teacher is.

So both can work, but the original sentence is the clearer and more standard way to say that they are patient to the same degree.

No, not if you mean my teacher.

You need the full noun phrase:

que mi profesora

Without mi, the meaning changes, and que profesora by itself would not sound right in this sentence.

So when comparing two people, you keep the determiner with the second noun phrase:

• que mi profesora
• que la profesora
• que una profesora

depending on what you mean.