Of course not, and it's a bit disingenuous to suggest that's what the parent believes.
It's pretty far fetched to suggest that an employer has a legitimate business purpose in collecting cookies for their employees' personal accounts that they happen to access on their work computer.
However, I think it's pretty easy to make the case that there's a legit business purpose in monitoring employee email (that is, email sent or received through an employer-issued email address), or at least in having access to it.
I can assure you that you won't have access to Gmail, or any other messaging service, when you work for a financial institutions.
The reason being that all official bank communications need to be stored and there are strict regulatory requirements that mandate that.
That does not mean that they have the right to read your email (at least in most European countries). Exceptions apply if dodgy dealings are suspected, but that's quite restricted and limited.
In any case and even with a relatively lenient internet policy any service, which can be used to exchange messages will be blocked by a bank.
...they also have access to gmail and a thousand other services, no?
> People have rights, even while working for an employer. They are employees, they are not slaves.
being asked to use gmail (etc) instead of company systems for your personal stuff doesn't 'make you a slave'.