Well musing doesn't in any way work for steemit, like all Dapps on the steem blockchain, they were created by independent units who saw that they could build something great on the steem blockchain.
Steemit and musing are under completely different management, the only thing is that they're both built on top of the steem blockchain and everything that happens on musing also happens on steemit.
In a way just like busy.org, musing is kinda like another user interface for steemit, except it was created to serve a different purpose, and your access to other parts of steemit like the trending page and hot page or people's wallet are limited or non-existent.
Steem is the name of the blockchain that Musing and a host of other applications use as their backend (*). Steemit Inc is the company that built the blockchain, which it does not own, however. Steemit.com is the first web interface into the blockchain. As you might have guessed Steemit.com was developed by Steemit Inc. Steem is owned collectively by all the people who own Steem Power. Each SP represents a small piece of ownership and control over the network.
*) In software architecture, backend means a system where the data are stored and front end is the presentation layer. A front end such as a web user interface like Steemit.com allows the user to see what's stored on Steem which is the backend and manipulate it.
Musing is independent of Steemit Inc but the latter has been loaned (delegated) a very large amount of Steem Power to Musing so that Musing could upvote its users to fuel its growth. Steemit Inc has done this because the people of Steemit Inc know that the value of the entire network depends on the success of the apps using it. They consider Musing to be a promising one.
First of all the existence of musing.io is not what we can term as working for steemit, musing.io is a decentralized app functioning via the steemit blockchain, in a way it's independent because it fuctions on it's own interphase, has it's own curators, it's own separate type of activities which is basically asking and answering of questions, just like decentralised apps like Tasteem which basically deals with food reviews and functions via the steemit blockchain and also reward it's user with steemit.
If you just realised a dapp just stop functioning via the steemit blockchain (Dlive) and relocated to another blockchain called Lino, this means it no longer had ties with steemit, or fuctioning under the steemit blockchain
Musing.io doesn't stand alone on an blockchain technology neither does it work for steemit, the correct term to use will be musing.io is functioning or operational under the steem blockchain.
Musing is aDapp that has been created on the Steem block chain. They are decentralized but only linked because of the use of the block chain.
They don't work for Steemit but work along side it using the Steem delegation that they were fortunate enough to receive. musing was created by developers independent of Steemit Inc. Musing adds value to the block chain and will draw new users when the SMT's are launched.
You have to log in using the address musing.io but is connected to Steemit like Steem Plus and Busy.org are.