Just what it says in the title. Are there any other rival brands of the Knight Bus, or just one? Is it the only weird bus thingy like that?
It probably is unique (at least in Britain).
The Knight Bus is a relatively modern invention in wizarding society, which sometimes (though it will rarely admit it) takes ideas from the Muggle world. The need for some form of transportation that could be used safely and discreetly by the underage or the infirm had been felt for a while and many suggestions had been made (sidecars on taxi-style broomsticks, carrying baskets slung under Thestrals) all of them vetoed by the Ministry. Finally, Minister for Magic Dugald McPhail hit upon the idea of imitating the Muggles’ relatively new ‘bus service’ and in 1865, the Knight Bus hit the streets.
Note the singular: only one Knight Bus hit the streets in 1865. However, this doesn't preclude the possibility that more Knight Buses were commissioned in the next 150 years.
Perhaps an even more important piece of evidence is that during Harry's short journey on the Knight Bus in Prisoner of Azkaban, it stops at a lot of different places in quick succession:
"Where are we, Ern? Somewhere in Wales?" [...] "We'll be in Abergavenny in a minute." [...] the bus moved abruptly from Anglesea to Aberdeen
If there was more than one Knight Bus in Britain, one would expect them to be regionally separated. The fact that this one bus travels between Wales, Scotland, and London within a span of minutes suggests that it's the only one in the country.
As usual in the Harry Potter series, though, everything is very UK-centric. It was the British Ministry for Magic that commissioned the Knight Bus, and the bus we see still seems to be confined to Britain. Whether other countries followed suit in developing magical bus services is unknown, but certainly plausible.
Given a reasonable estimate of the wizarding population of Great Britain being three to ten thousand, the availability of more convenient methods of travel, and the relative efficiency of the Knight Bus compared to Muggle public transport, it seems unlikely that more than one would be needed.
Many Muggle towns of that size (at least in my part of the world) have no public transport whatsoever.