In a prototype & R&D lab where I formerly worked, there were 'standard' lab passwords for things like root, administrative access to consoles, switches, etc. These are simple, easy to remember, and shared verbally with anyone who needed them. In general, if you could physically get into the lab, you were authorized to have these passwords.
In the manufacturing facility, new systems were built and configured for customers. The customer got to choose all the passwords, and they were printed on a set of forms that were attached to the rack with the systems. Remote access was provided as required, and the passwords were sent in an e-mail, or given over the phone. It was fully expected that the customer would change these passwords as soon as the system was delivered to them.
For the IT & Production labs, almost no one had root access. Almost everyone did have sudo access with somewhere between no limits and only the ability to mount virtual filesystems...depending on the person and the system. It was very rare to get sudo access to launch a shell as root. This left a very clear log trail of all the commands you ran as root. That log was used to tar & feather more than one person over the years.
At a help desk / support role I had many years ago, each tool expert picked their own administrative passwords. These were recorded in an envelop that was locked in a safe in the machine room. If someone needed admin access, they could open the envelop, read the password, and note in the log that they knew the password and then re-seal the password in the envelop. It was up to the tool owner to decide if the password needed to be changed. This system was used for more than 5 years...and in one case actually helped the project to survive the "bus test" (heart attack) for one team member.
Different standards for different kinds of systems and labs. That is reasonable. I find that when passwords need to be shard, it is best if the password is simple, short, and communicated verbally (either in person or over the phone). I find that the only password that should never be shared is the one for my personal account. Any root/admin/tool specific passwords should be backed up in at least one other head...if not recorded in some manner.