The abaya "cloak" (colloquially and more commonly, Arabic: عباية ʿabāyah , especially in Literary Arabic: عباءة ʿabāʾah ; plural عبايات ʿabāyāt , عباءات ʿabāʾāt ), sometimes also called an aba, is a simple, loose over-garment, essentially a robe-like dress, worn by some women in parts of the Muslim world including in North Africa and the Arabian Peninsula. Traditional abayat are black and may be either a large square of fabric draped from the shoulders or head or a long kaftan.
Islam preaches inner and outer modesty, and as human beings, we feel more safe and secure when we are covered up. This is why all religions somehow promote covering up and not being exposed for onlookers to see.
If a non-Muslim wishes to dress modestly, then of course that will benefit her. If she wishes to dress like a nun, or as Muslims, then that is also her choice, and it would not be considered as being 'inappropriate'.
In the case of a Muslim woman, or non-Muslim, wanting to, for example, enter into a holy shrine, there is a requirement that she does wear an abayah/chador. It is not Islamically mandated, but just out of respect.
And Allah knows best.
For the most part, we don't have strong historical evidence about the details of what women from the Ahl al-Bayt (A) were wearing, apart from a few situations where there was a reason to mention a specific fabric or something.
Anyway, without being an expert in the history of clothing, it seems that in some places, the tendency to wear black is pre-modern (and I suspect due to convenience) and in some places it is a product of modernity. When discussing today's "Islamic fashions", it is worth keeping in mind that what we have access to today is largely due to industrialization - for instance, the production of synthetic fabrics, cheap stretchy socks, and mass-produced garments - and people in the past did not have these things. Also, the concept of global "Islamic fashion" (often epitomized by the black Saudi-style abayah) is relatively new and probably due to globalization; up until recent decades, Muslims seemed to tend to wear their own regional styles more.
(There are, however, some interesting books about the history of clothing in the Muslim world, if it is a subject of interest, as well as some early photographs)
To my knowledge, in places like Iran, in the past, it was more common to wear coloured chadors, but in the 20th century, black became the popular colour.
Anyway, shari'ah in and of itself does not regulate what colour you have to wear. However, a general principle is that of 'urf; that is, one should dress in such a way which is considered normal in the area and does not attract attention (unless there is a clear reason to do otherwise, for instance, everyone may be wearing a bikini on the beach, but that doesn't mean it's appropriate to wear a bikini). So colours are one of the matters of 'urf; in some places, it might be normal and appropriate to wear bright colours, and in other places, such as Qom, it might attract a lot of attention or be an inappropriate form of social signalling. Non-Muslm countries also tend to vary widely in their 'urf regarding the colours people wear so it is really left to one's individual judgment, or you could ask people around you if you are concerned.
Anyway, as is said, Allah is beautiful and loves beauty, and Allah created all the colours and did not make any of them inherently taboo.*
(*apart from cases where wearing certain colours is specifically considered makruh, this is not the main subject here)
Here is a link to an extensive document on the details of the fiqh of hijab:
A woman in presence of non-mahram must cover all her body except face and hands(wrists down) -even feet- with loose clothes so that the body shape shall not be seen underneath the clothes.
So Abayah or Chador is not mandatory.
The colors of the cloth of the female in front of the non Mahram men should not be extra attractive even if it covers her body as full Hijab. Extra attractive colors must be avoided.
Any dress which fulfills the conditions of Hijab is allowed as far as it covers the whole body except the face and the hands and it is not tight and not transparent.