No. You Slack user and profile is always part of the individual Slack team / workspace. Slack does not support global users.
As I understand that is part of Slack's security architecture, where it's not possible to see other Slack teams unless you have been explicitly invited.
There's a menu option in the dropdown on the top left (the label is the current team name). Open the menu and select "Sign in to another team...". I don't know if something has changed since my original question, but I'm now able to sign in to multiple teams simultaneously using different accounts. Previously, it didn't even prompt me to login; it just tried to re-use the credentials of the first team and failed.
Best Answer
From Slack support: