Usage and origin of "sister" in expressions like "sister company, sister ship, sister site" etc

'Brother company' - or 'brother (anything)' - would almost certainly be considered incorrect (in English). There's no logical reason why it should be incorrect, only historical.

You're right that the first usage of "sister (object)" was probably for ships. I can't find anything earlier.

We use 'she' to refer to something which we have so much affection for that it almost seems rude to refer to it as 'it'. A common usage today is people referring to their cars as if they were female.

When this usage first arose, men dominated society and certainly dominated both ship-building and seafaring. Men, generally, have an affinity for women and so its only natural that they would start to use the pronoun 'she' for something that they were similarly fond of. For long periods at sea, men might feel nurtured by their ship - a quality that we generally associate more with women.

It's possible the usage directly transferred from ships to companies but I see this as quite speculative. It's a nice idea, though.

The first proper companies were formed to finance sea voyages. Each company was formed for a single ship voyage and dissolved at the end. Since the ship basically was the company, the usage of "she" for the ship transferred to the company itself. Thus we now have sister, daughter, etc. companies.

The user speedwell2

This is probably also at least partially a cultural thing. Other cultures do not necessarily want feminine qualities associated with their corporations. They would prefer for their companies to embody more traditionally masculine characteristics like strength and solidity. Supposedly in China companies are more often referred to as masculine, for example.

With a growing pressure to avoid using gender-specific language, I expect the usage of phrases such as 'sister company' to decline in favour of more neutral terms such as 'subsidiary'.

It happens that "sister" (and "mother" and "daughter") are used for relationships between various inanimate entities - ships, companies, schools, monasteries, languages - and not "brother" or "father" or "son". This is simply a fact about English, with no obvious explanation.

I'm dubious that this has anything at all to do with the use of "she" for ships and countries, but I may be wrong.

The answer as to why the term brother is never used in context with belonging to the same group, class, or organisation could lie in biology.

Women bear children, they are able to generate, and create new life. Likewise, if a company expands and creates (procreates) a new company, that "baby" company is related to its parent.

Sister cell

From Text-book of Botany: Morphological and Physiological, 1875, an illustration showing the protruding cell-wall containing the daughter-cells.

B the inner lamella of the mother-cell-wall which has entirely escaped (greatly enlarged).

SE Biology

During the days when philosophers used to debate, they tended to regard reproduction as a feminine trait. So naturally organisms/cells capable of producing offspring are also given a feminine trait. The parent cell is often called the mother cell, and the daughter cells are so named because they eventually become mother cell themselves.

It is no coincidence that a sister company is also called (less so today) a daughter company

Thus a sister site can be created or set up, and the main site is said to be the parent. Radio and TV stations, own sister stations and channels. This dispels the concept that the feminine pronoun is used as a term of affection. In a historical context, there is nothing cosy or affectionate about being ‘owned’ by a larger company, although it is in the parent company's best interest that their "daughter company" is equally successful.

sister company
A company which is owned by the same parent company as another company. One parent company can have one or many subsidiaries, which all are sister companies to each other. Business Dictionary

Sister Ships

An excerpt from The Naval Chronicle (including the biographical history of the Royal Navy of the United Kingdom), written in 1813, shows that the term sister ship was firmly established by the 19th century, also note the feminine pronoun used.

She is the sister ship [A French ship named Andromaque], in every respect, to the Weser; for their keels were laid down on the same day; they were launched the same day; sailed the same day ; were dismasted on the same day; were brought into Plymouth on the same day; and had a similar number of men, and weight of metal. The capture of these two vessels may perhaps be considered as doing Buonaparte a favour, inasmuch as it may spare him hereafter many unpleasant recollections attached to their names.

The excerpt illustrates perfectly the meaning of sister ship in that period. Today, the International Maritime Organization includes the following characteristics:

• A sister ship is a ship built by the same yard from the same plans.
• The acceptable deviation of lightship displacement should be between 1 and 2% of the lightship displacement of the lead ship, depending on the length of the ship.
  _Wikipedia

The earliest example of sister ship I found on Google books, was its plural form, in a French-English Naval dictionary, titled Vocabulaire des termes de marine, printed in 1799.

All of which may appear to contradict my earlier statement, but I don't think it does. The term sister is derived from biology, and in the shipping industry, it refers to a ship built at a later date but following the same design and specifications as the "older sister ship". The parentage (mentioned by @BaconBits in the comments below) is the same, therefore any ship ‘created’ in the same yard, with the same hull, a similar weight, and equipment etc. is, in a figurative sense, a sister.

(Answering your question "could brother be used instead?") The OED says:

brother, n. IV. 10. A thing perceived as resembling, or having a close connection to, another or others.

with citations including:

a1475   (▸a1376)   Langland Piers Plowman (Harl. 875) (1867) A. ii. l. 141 (MED), Feire speche þat is feiþles is falsnes broþer.

1802   Wordsworth in Wordsworth & S. T. Coleridge Lyrical Ballads (ed. 2) II. 124   That April morn, Of this the very brother.

1830   Tennyson Isabel in Poems 8   A clear stream flowing with a muddy one, Till in its onward current it absorbs..The vexéd eddies of its wayward brother.

1911   Polit. Sci. Q. 26 164   In the United States, the telephone has grown to be the big brother of the telegraph.

1978   J. Maxwell America's Fascinating Indian Heritage iii. 97/2   The stickball game the Southeastern tribesmen aptly called the little brother of war.

The use of "sister" for a related item has been common since the 13th century, and seems to be independent of the gender of the noun.

OED

Sister

IV. Something which is closely related to another thing.

9. Something having a close kinship or relationship to another; something belonging to the same class or group. Sometimes personified.

See also appositive uses with the sense ‘fellow’.

a. With reference to an immaterial thing, esp. a quality, emotion, or behavioural trait.

a1225 (▸c1200) Vices & Virtues (1888) 29 Of rihte ȝeleaue. All ðat hire suster, ðe rihte ȝeleaue, hire seiȝeð, all hie [sc. hope] hit fastliche hopeð1. [Of True Faith: All that her sister, true faith says to her, she sincerely trusts it.[2]]

b. With reference to a material thing, place, etc.

a1500 (▸c1477) T. Norton Ordinal of Alchemy (BL Add.) (1975) l. 2320 (MED) Quycsyluere..wil nevir cleve to a thing But to metal of oone kind or odyre, For there he fyndith sustir or brodyre. [Quicksilver (mercury)... will not stick to anything except to a metal of one kind or another, for thatis where he will find a sister or brother.]

1580 T. Crewe tr. G. Meurier Nosegay of Morall Philos. sig. F3 Q. Who is sister to death? A. Sleepe.

2005 Independent (Nexis) 5 Feb. 45 The drink incorporates creme de mure (blackberry liqueur), the somewhat sweeter sister of creme de cassis (blackcurrant liqueur).

Sister, in this [attributive] sense thus pre-dates the nautical influence.

†10. Nautical. Any of various items of tackle having two or more matching components.

Obsolete.

1410–12 in N. H. Nicolas Hist. Royal Navy (1847) II. 475 Un rakke ove ii. sustres, un trusp'aill ove ii. sustres, ii. slenges, un trusse, un canon.

NB: by this time, (i) nouns had lost their gender. (ii) The Old English noun "scip" (ship) was not feminine - it was neuter https://bosworthtoller.com/57769

1 Of feste hope. HIER after cumþ an oðer hali mihte ðe is iclepedfirma spes, þat is, fast hope to godalmihti. All ðat hire suster, ðe rihte ȝeleaue, hire seiȝeð, all hie hit fastliche hopeð.

[2] my translation