Note: I spent a few years studying the Swahili language in college, and branched out into an African Studies minor focusing on early East African history having become fascinated with the culture of the Swahili coast. A lot of my teachers were Tanzanian, so I have a slightly skewed picture. Much of this is from memory, but I'll throw in some links to newer scholarship.

The Swahili language is a really interesting thing... it grew out of local, coastal people in East Africa whose culture developed alongside Arab traders that settled in the area. It's important to understand that the language is Bantu in origin, but has been massively influenced by Arabic (and, much later, in Swahili history, English). Kind of like how English is a Germanic language, but something like 40% of its vocabulary is of non-Germanic origin, Swahili developed on a trading coast with lots of contact with non-locals, so much of its vocabulary derives from non-Bantu origins, primarily Arabic, followed by English, but also Hindi, Persian, and Portugese.

The name Swahili derives from Arabic: sahil (coast) or sawahil (plural), with sawahili (of the coast). The in-language term for the language itself is Kiswahili, and people who natively speak it are known as Waswahili.

Persians settled along the Swahili coast in the early medieval period. Sources vary, and the early scholarship recording this is kind of muddy, but it could have been as early as the late 900s. It's well established that the Kilwa Sultanate was up and running by the 1200s on the island of Kilwa off what's now the Tansanian coast, south of the Zanzibari islands. At some point later (I honestly don't remember exactly when, sometime after the 1500s for sure) Omani influence became stronger than Persian influence. The Persian (Shirazi) and Omani settlers all along this way intermarried with local Islamic converts and created a unique culture that was partly Persian, a lot Arabic, and mostly African.

Local Swahili people considered themselves (and many still do!) Arabs, and they established a pretty hefty trading empire as far north as what's now Somalia and as far south as Madagascar, with coastal ports and cities dotting the coastline. They traded with folks from the interior, and set up pretty extensive trading networks all across the Indian ocean. Some of the earliest written records in the Swahili language are written in the Arabic script and collected in the Portugese colony of Goa, on the western coast of India, as trade letters. During all this shifting and development, the center of trade and influence moved from Kilwa to Zanzibar (and in fact, no one lives on Kilwa any longer, really.)

So Swahili became a lingua franca already during the medieval period. There's a lot of linguistic scholarship around various dialects of the Swahili language continuum, but the fact that there's a bunch of dialects now reflects that the "Swahili as she is spoke" back in the day was already somewhat different.

The Omani Empire (who were kind of clients to the Portugese in the 1500s but eventually threw them off in the later parts of the colonial era) eventually fractured and was actually headquartered on Zanzibar for the very end of its era in the mid 1800s as well. Given the already established "We are Arabs" culture, this is pretty much why the modern view that proper Swahili is from Zanzibar - kind of the last holdout of the island settlements of the Shirazi/Kilwa days.

When European colonial powers established tight control in the 1800s (I focused on medieval history - the Shirazi/Arab period, basically, so now I'm in a territory that I'm terribly vague on dates - from this point forward, my history is going to be... loose), various decisions were made to govern the locals by colonial governors. What became Kenya was under one governor, and what became Tanzania was under another under British rule. Germany had some say so during this period for a while as well, but lost control after World War 1, where it defaulted back to the British. This further divided language flow. When Kenya and Tanzania won independence in the mid-1900s (after the colonial powers and the UN were trying to pick up the pieces of fractured colonial rule post-WW1 and the fallout from that), Swahili was still the main lingua franca.

Tanzania teaches primary school in Swahili. Most people who don't live in the coastal areas speak a different language at home, and pick up Swahili in school or through trade/tv/radio/etc. English is the instruction language of higher education. Kenya (at least up 'til when I was in college in the 90s) was similar, except primary school instruction is (was?) in English. Coastal areas and some of the bigger cities have a lot of Swahili speakers.

So... basically the language started changing almost as soon as it was spreading. It's not a new process at all. I'm not up to speed on newer developments at all, but I hope this can put a little perspective into things.

Sources:

https://www.amazon.com/Swahili-speaking-Peoples-Zanzibar-African-Coast/dp/0853020345 (This was one of my references when I was doing my term paper)

https://www.amazon.com/dp/B06WLHVPSQ/ref=dp-kindle-redirect?_encoding=UTF8&btkr=1 (likewise)

https://books.google.com/books?id=lGwnDwAAQBAJ