I successfully uninstalled Opera Browser Assistant and it remained uninstalled after my last Opera update from within Opera itself.

Here I will share information about it, so others may benefit from it.

Uninstalling Opera while keeping your personal data and installing Opera again using the Offline Opera Installer no longer works te remove Opera Browser Assistant.
Apparently the installer now detects you had Opera Browser Assistant installed and installs it again.

Here is how I removed Opera Browser Assistant:

• Uninstall Opera without deleting your personal user data.
• Download and run the Offline Opera Installer.
• Do not start the installation yet, but open Explorer and go to the folder:

%localappdata%\Temp\.opera\Opera Installer Temp

• In this folder go to the folder containing the setup files. The folder is named like this:

opera_package_<timestamp>

• Delete the file named "extra_apps".
  In my case Opera Browser Assistant was the only component listed in this file. You may want to edit this file if you have more components listed in this file.
  Alternatively you can remove the folder "assistant" and leave the "extra_apps" file intact.
• You can now close Explorer and continue with Opera Installer.
• Set your preferred options in the Opera Installer and install Opera.

After this, new Opera versions downloaded by the Opera Update feature, no longer seem to include the Opera Browser Assistant package.

Plenty of updates further and still no sign of Opera Browser Assistant

When in private browsing mode and clicking for example on a link to an atom feed, Opera crashes. In normal mode it works OK.

The RRS feed link on the homepage op Notepad++ also makes Opera crash in private browsing mode. Also when I paste the link in a new tab. The link is:
https://notepad-plus-plus.org/index.xml

Is this a known issue?

(Opera version 88.0.4412.27)

Tested Opera 89.0.4447.48 64-bit. The issue is solved.

Seems to be fixed now in version 90.0.4480.80.

Tested Opera 89.0.4447.48 64-bit. The issue is solved.

Still crashing in version 89.0.4447.38.

Same for Opera 88.0.4412.74

Same for Opera 88.0.4412.53

Same for Opera 88.0.4412.40

The atom feeds of this forum and of Wikipedia doesn't seem to be supported by Opera. When I click the feed link, it says:
This XML file does not appear to have any style information associated with it. The document tree is shown below.

What's up?

(Opera version: 88.0.4412.27)

When in private browsing mode and clicking for example on a link to an atom feed, Opera crashes. In normal mode it works OK.

The RRS feed link on the homepage op Notepad++ also makes Opera crash in private browsing mode. Also when I paste the link in a new tab. The link is:
https://notepad-plus-plus.org/index.xml

Is this a known issue?

(Opera version 88.0.4412.27)

Plenty of updates further and still no sign of Opera Browser Assistant

Some technical notes concerning Opera's installation process

Directly after starting Opera's setup program, the following folder is created:

%localappdata%\Temp\.opera\Opera Installer Temp\opera_package_<timestamp>

The following files are put in this folder:

• opera_package
  7-zip file containing Opera program files including the file assistant_package.
  assistant_package is a self-extracting zip file containing Opera Browser Assistant installation files.
• assistant\assistant_installer.exe
  assistant\browser_assistant.exe
  assistant\files_list
  assistant\mojo_core.dll
  Opera Browser Assistant installation files
• extra_apps
  This file contains the text "assistant:assistant".
  The setup program creates this file after successfully launching:

assistant_installer.exe --version

• ready
  Empty file probably indicating preparations are finished.

After you have set your preferred options in the setup program and started the installation, Opera is installed. To install Opera Browser Assistant, the setup program uses the following command line (may deviate depending on chosen options):

assistant_installer.exe --installfolder=<Opera install folder>\assistant --copyonly=0 --allusers=[0/1]