The reason for this is because until the early 1700’s English spelling wasn’t standardized. Multiple dictionaries published at that time would show the spelling conventions on more of a regional level in the States and across the British Empire. Two works, Samuel Johnson’s A Dictionary of the English Language, from 1755 and Noah Webster’s An American Dictionary of the English Language from 1828 are the basis for English and American spelling, respectively. Back then, Webster was even a proponent of Spelling Reform, which worked to ensure spelling in the English language was more consistent and phonetic.
Today, Americans use the American standards for spelling, color, labor, modeling, center, liter (even though they don’t use that either) neighbor, and fiber, and the Brits and other countries that were formerly members of the British Empire, Canada, India, Australia, New Zealand, etc, use the British form*. Officially, Canada uses the British standards, but pockets of Canadians don’t bother with adhering to those conventions. Not that it bothers me, or anything. Institutions like the government, media networks, publishing houses, and official bodies will use the British standards, but it’s not like there’s a language police who can monitor whether people are putting u’s in the right places. Oh, wait, there is. Well, those police don’t really concern themselves with the u’s in English.
Many words in English which end with -re, originate in the French, Latin, or Greek languages. Fibre is derived from the Latin fibra, meaning filament, or entrail (ew) and was incorporated into English from Middle French around the 16th century. When English became more common, the American standards changed the -re suffix to -er. Words like lustre, sombre, theatre are luster, somber, and theater in American English. The e before the r is kept in place in American-derived forms of nouns and verbs, like the noun fiber.
Though the spelling is different, both words mean textile fibre as well as dietary fibre in their respective spelling standards. There is not two spellings for two definitions. Remember that in Milwaukee, asking someone if they’re getting enough fibre means the same thing as in Manchester.
*colour, labour, modelling, centre, litre, neighbour, and fibre, in case you weren’t paying attention in English classes.