Ubuntu's do-release-upgrade command upgrades the operating system to the latest release. What is Debian's way or tool for the same purpose (to upgrade to the latest stable release)?
Debian does not provide a single command to upgrade the OS to a new release. The Release Notes for each release include upgrade instructions for supported hardware architectures.
You can find release notes for all Debian releases via the Debian Releases page.
For example, to upgrade a 64-bit PC from stretch to buster, follow the instructions in Chapter 4. Upgrades from Debian 9 (stretch) under Debian 10 -- Release Notes for Debian 10 (buster), 64-bit PC.
You should always be able to find the release notes for the current stable release at https://www.debian.org/releases/stable/releasenotes.
Although upgrading a Debian release from "oldstable" to stable is usually painless, it's important to follow the Release Notes because the OS can differ from release to release in ways that could affect your specific installation.
The Release Notes also contain information and tips about changes in the new release that can save considerable time and effort.
For example, the upgrade process for some previous releases recommended the use of
aptitude for the upgrade.
For upgrades from stretch to buster, the
apt tool is recommended instead of
aptitude is suggested for resolution of some problems after the upgrade.)
Officially, there's no tool - you're supposed to take care about it on your own: see https://www.debian.org/releases/stable/i386/release-notes/ch-upgrading.html, for example.
Basically, it revolves around changing/adding URLs to repository for newer version and upgrading packages, sorting out any trouble that might come. You can see in the guide that it's not really convenient. This is why many distributions don't provide way to upgrade between releases, or strongly discourage it - there's a lot to care about (and also why it's often good to use LTS or rolling release distributions).
There might be an unofficial tool to automate the process in Debian, but I would rather follow the documentation in that respect.