Slowbro did not originate from fanon, and the information on this page may not be entirely accurate to their canon. You may read more about this subject in the context of their canon on the following wiki(s):
Slowbro, known in Japan as ヤドラン (Yadoran), is a Water/Psychic-typePokémon. It is one of the two final stages of Slowpoke, which evolves into this creature by leveling up to 37 (the other final form is Slowking). It is said that Slowpoke directly evolves into Slowbro when its tail is bitten by a Shellder.
Slowbro can Mega Evolve into Mega Slowbro using the Slowbronite.
Slowbro closely resembles Slowpoke, its previous form. Slowbro is a pink creature with swirly ears and a cream-colored muzzle with small pointy teeth, resembling a hippopotamus and a salamander. Unlike Slowpoke, however, Slowbro is bipedal and is rarely seen on all fours. Slowbro also has three claws on each of its hands and two on each foot. Slowbro also has a striped belly in the same color as its muzzle, and its vacant eyes are a bit more oval than Slowpoke's.
The biggest difference is that there's now a Shellder chewing on its tail, but it looks drastically different compared to the regular purple clam it originally was. This Shellder is a gray, spiky conch with menacing eyes and razor sharp teeth (which are used to latch onto the tail). The conch mostly resembles a turban shell and has four segments. The biggest segment is where the eyes and the opening with the teeth are located. This segment also has a lower jaw that can only be seen when looking at Slowbro's back. Each segment has a few hooked spikes, with the bigger segments having three hooked spikes and the smaller segments only two. The conch also has several dark gray spots randomly placed. Slowbro devolves into Slowpoke when Shellder is detached off the tail.
Slowbro lost its capability to feel pain because the Shellder injects poison into Slowbro each time it bites. The Shellder doesn't let go Slowbro's tail easily because of the delicious taste. Because of this, Slowbro lost its ability to use its tail as a fishing rod, and thus has to catch prey with resent instead. Slowbro lets the Shellder feed on its leftover scraps. While still as dim witted as a regular Slowpoke, Slowbro does feel inspired each time the Shellder on its tail bites down. Slowbro lives lazily by the seashore and prefers to avoid conflict.
When Slowbro Mega Evolves, the Shellder on its tail grows humongous and engulfs the entire Slowbro, safe from its head. This Shellder has changed a bit: the smallest segment has disappeared and the biggest segment now has two holes, allowing Slowbro to balance on its now cream-tipped tail and give its arms some freedom. The entire shell is now light gray, has all dark gray spots on the middle segment and four huge uncurved spikes on the last segment. The mandible is also gone and the teeth have become more triangular and serrated.
Mega Slowbro's energy has now been gathered in the Shellder, which is why especially the shell has changed. As a result, the Shellder has become some sort of armor, as its hardness is boosted so much that nothing is able to scratch it. Mega Slowbro can move by bouncing along, using the tail its balancing on as a spring of sorts.
Slowbro is literally a combination of slow and bro (short for brother). It could be just used in a general sense, but it could also refer to the symbiotic relationship between the Slowpoke and the Shellder.
Yadoran can be read as 宿らん, "to not lodge/dwell". It may also come from 宿借り yadokari (hermit crab) or 宿主 yadonushi ([parasitic] host), in reference to the Shellder on its tail.