I work in cybersecurity for a large company, which also uses the MS Authenticator app on personal phones (I have it on mine). I do get the whole "Microsoft bad" knee-jerk reaction. I'm typing this from my personal system, running Arch Linux after accepting the difficulties of gaming on Linux because I sure as fuck don't want to deal with Microsoft's crap in Windows 11. That said, I think you're picking the wrong hill to die on here.
In this day and age, Two Factor Authentication (2FA) is part of Security 101. So, you're going to be asked to do something to have 2FA working on your account. And oddly enough, one of the reasons that the company is asking you to install it on your own phone is that many people really hate fiddling with multiple phones (that's the real alternative). There was a time, not all that long ago, where people were screaming for more BYOD. Now that it can be done reasonably securely, companies have gone "all in" on it. It's much cheaper and easier than a lot of the alternatives. I'd love to convince my company to switch over to Yubikeys or the like. As good as push authentication is, it is still vulnerable to social engineering and notification exhaustion attacks. But, like everything in security, it's a trade off between convenience, cost and security. So, that higher level of security is only used for accessing secure enclaves where highly sensitive data is kept.
As for the "why do they pick only this app", it's likely some combination of picking a perceived more secure option and "picking the easiest path". For all the shit Microsoft gets (and they deserve a lot of it), the authenticator app is actually one of the better things they have done. SMS and apps like Duo or other Time based One Time Password (TOTP) solutions, can be ok for 2FA. But, they have a well known weakness around social engineering. And while Microsoft's "type this number" system is only marginally better, it creates one more hurdle for the attacker to get over with the user. As a network defender, the biggest vulnerability we deal with is the interface between the chair and the keyboard. The network would be so much more secure if I could just get rid of all the damned users. But, management insists on letting people actually use their computers, so we need to find a balance where users have as many chances as is practical to remember us saying "IT will never ask you to do this!" And that extra step of typing in the number from the screen is putting one more roadblock in the way of people just blinding giving up their credentials. It's a more active thing for the user to do and may mean they turn their critical thinking skills on just long enough to stop the attack. I will agree that this is a dubious justification, but network defenders really are in a state of throwing anything they can at this problem.
Along with that extra security step, there's probably a bit of laziness involved in picking the Microsoft option. Your company picked O365 for productivity software. While yes, "Microsoft bad" the fact is they won the productivity suite war long, long ago. Management won't give a shit about some sort of ideological rejection of Microsoft. As much as some groups may dislike it, the world runs on Microsoft Office. And Microsoft is the king of making IT's job a lot easier if IT just picks "the Microsoft way". This is at the heart of Extend, Embrace, Extinguish. Once a company picks Microsoft for anything, it becomes much easier to just pick Microsoft for everything. While I haven't personally set up O365 authentication, I'm willing to bet that this is also the case here. Microsoft wants IT teams to pick Microsoft and will make their UIs even worse for IT teams trying to pick "not Microsoft". From the perspective of IT, you wanting to do something else creates extra work for them. If your justification is "Microsoft bad", they are going to tell you to go get fucked. Sure, some of them might agree with you. I spent more than a decade as a Windows sysadmin and even I hate Microsoft. But being asked to stand up and support a whole bunch because of shit for one user's unwillingness to use a Microsoft app, that's gonna be a "no". You're going to need a real business justification to go with that.
That takes us to the privacy question. And I'll admit I don't have solid answers here. On Android, the app asks for permissions to "Camera", "Files and Media" and "Location". I personally have all three of these set to "Do Not Allow". I've not had any issues with the authentication working; so, I suspect none of these permissions are actually required. I have no idea what the iOS version of the app requires. So, YMMV. With no other permissions, the ability of the app to spy on me is pretty limited. Sure, it might have some sooper sekret squirrel stuff buried in it. But, if that is your threat model, and you are not an activist in an authoritarian country or a journalist, you really need to get some perspective. No one, not even Microsoft is trying that hard to figure out the porn you are watching on your phone. Microsoft tracking where you log in to your work from is not all that important of information. And it's really darned useful for cyber security teams trying to keep attackers out of the network.
So ya, this is really not a battle worth picking. It may be that they have picked this app simply because "no one ever got fired for picking Microsoft". But, you are also trying to fight IT simplifying their processes for no real reason. The impetus isn't really on IT to demonstrate why they picked this app. It is a secure way to do 2FA and they likely have a lot of time, effort and money wrapped up in supporting this solution. But, you want to be a special snowflake because "Microsoft bad". Ya, fuck right off with that shit. Unless you are going to take the time to reverse engineer the app and show why the company shouldn't pick it, you're just being a whiny pain in the arse. Install the app, remove it's permissions and move on with life. Or, throw a fit and have the joys of dealing with two phones. Trust me, after a year or so of that, the MS Authenticator app on your personal phone will feel like a hell of a lot better idea.