One of the problem's with MDM software is that corps want you to login and use your personal phone, I guess to save costs, and to make it easy for you to do work out of your regular business hours.
If a company asked me to use MDM software and set themselves up as a device owner on a phone I purchased and used every day my answer is: hell no
If they want that, they can buy me a phone, and pay for the mobile/data plan. I've worked places that have done this, having 2 phones is a pain, but you only use the corp one at work or if you're oncall.
BYOD support without having everything managed by the company is a pain point Apple and Google are trying to solve.
For Chrome, it will perform a very intrusive popup whenever you log into an extra Google account to get you to use a different profile. If you say yes, that new profile will be governed by the administrators without them gaining access to the entire Chrome browser.
For Android, there are 'Work Profiles'[0], however I haven't tried this and I wouldn't be surprised if it breaks fundamental parts of Android and/or it's disabled on certain OEM Android makers.
For iOS, User Enrollment[1] is a thing.
The main problems I see with these solutions is that they add a lot of complexity to MDM configurations, so chances are the organization will either go without MDM, or ask you to set up your device under full MDM. Under the second scenario I would suggest purchasing an extra phone just for work - this also helps with the possibility of an internal investigation, or even subpoena, asking for access to any phone with work data on it, as chances are they won't limit searches and data exports to data stored in your work profile.
At a previous company I declined the "perk" of the company paying for my phone plan, because it required giving them control over it. I was mostly worried about losing my phone number accidentally upon parting ways.
Nowadays, Android can have a Work profile that your company can control (and wipe, for example) that doesn't affect your personal stuff. It's actually convenient because you have a separate instance of Chrome, which is a good workaround for mobile Chrome not supporting multiple profiles inside the app like the desktop version does.
If a company asked me to use MDM software and set themselves up as a device owner on a phone I purchased and used every day my answer is: hell no
If they want that, they can buy me a phone, and pay for the mobile/data plan. I've worked places that have done this, having 2 phones is a pain, but you only use the corp one at work or if you're oncall.