A la meva mare li agrada aquest quadre de la ciutat.

Yes, they both refer to the same person, and this is completely normal in Catalan.

This is called clitic doubling:

• a la meva mare = to my mother
• li = to her / to my mother

Catalan often uses both:

• the full phrase for clarity or emphasis
• the pronoun because the verb structure expects it

So A la meva mare li agrada... is the natural Catalan way to say that my mother likes...

Because agradar works differently from English to like.

Literally, the structure is closer to:

• This painting is pleasing to my mother

So in the sentence:

• a la meva mare / li = the person who experiences the liking
• aquest quadre = the thing that is pleasing

That is why Catalan does not structure it like my mother likes this painting in a word-for-word way.

Because the verb agrees with aquest quadre, which is singular.

In sentences with agradar, the verb agrees with the thing liked, not with the person who likes it.

Here:

• aquest quadre = singular
• so the verb is agrada

Compare:

• Li agrada aquest quadre = She likes this painting
• Li agraden aquests quadres = She likes these paintings

The a marks the person affected by agradar.

With agradar, the person who likes something is introduced with a:

• A la meva mare li agrada...
• Al meu pare li agraden...

This is because that person functions grammatically like an indirect object.

In Catalan, possessives are usually used with the definite article:

• la meva mare = my mother
• el meu pare = my father
• la seva casa = his/her house

This is different from English, where you normally say just my mother without the.

So la meva mare is the standard Catalan form.

Because quadre is a masculine singular noun.

So the demonstrative must match it:

• aquest quadre = this painting/picture
• aquesta ciutat = this city

Catalan demonstratives agree in gender and number with the noun.

Quadre usually means painting or picture, especially an artwork.

Depending on context, it can refer to:

• a painting
• a framed picture
• sometimes a visual work in general

In this sentence, aquest quadre de la ciutat most naturally means:

• this painting/picture of the city

Here de la ciutat most likely means of the city or showing the city.

So:

• aquest quadre de la ciutat = this painting/picture of the city

In context, it could mean:

• a picture depicting the city
• a painting associated with the city

Most learners should understand it here as a picture of the city.

Yes. That is a perfectly normal sentence.

• Li agrada aquest quadre de la ciutat = She likes this painting of the city / My mother likes this painting of the city

But without a la meva mare, the listener has to get li from context:

• li = to him / her / you-formal

So adding a la meva mare makes it explicit who the sentence is about.

Yes, Catalan allows some flexibility in word order.

The given sentence:

• A la meva mare li agrada aquest quadre de la ciutat

is very natural, especially when introducing or emphasizing my mother.

You could also hear:

• Aquest quadre de la ciutat agrada a la meva mare
• Li agrada aquest quadre de la ciutat, a la meva mare

But the original version is one of the most common and natural ways to say it.

In standard Catalan, li is singular:

• to him
• to her
• to you (formal singular)

For plural, standard Catalan uses els:

• A les meves germanes els agrada aquest quadre = My sisters like this painting

So in your sentence, li matches the singular person la meva mare.

The verb would change to agree with the plural noun.

Singular:

• A la meva mare li agrada aquest quadre = My mother likes this painting

Plural:

• A la meva mare li agraden aquests quadres = My mother likes these paintings

What changes:

• agrada → agraden
• aquest quadre → aquests quadres

The person phrase a la meva mare li stays the same.

It is completely normal everyday Catalan.

Nothing in it sounds especially formal or literary. In fact, the structure with:

• A la meva mare
• li agrada

is exactly the kind of thing you hear all the time in natural speech.