Why does an environment-variable change in one terminal not appear in another?

Not necessarily. In PowerShell, the environment available to a session belongs to its process. A change made in that session affects that session, while another terminal session can have a different environment. [S1] The change is not automatically a live update to every terminal that is already open.

A useful picture is:

Parent process
├── PowerShell session A — its environment
│   └── Child process — inherits A's environment when started
└── PowerShell session B — its own environment

This is a simplified model: a child process inherits from its parent, but two separate terminal sessions do not thereby become one shared session. If terminals were started from different parents, their starting environments may differ too. This source describes PowerShell and Windows behavior; other shells and operating systems can have additional details. [S1]

Yes. In each of two PowerShell terminals, inspect the same variable name before making changes. Then set a clearly disposable name in the first terminal, inspect it there, and check that name in the second. For example, use GEEZ_SCOPE_TEST rather than the name of a setting your work depends on.

In PowerShell, these commands set, display, and remove that process environment variable:

$env:GEEZ_SCOPE_TEST = "session-A"
$env:GEEZ_SCOPE_TEST
Remove-Item Env:GEEZ_SCOPE_TEST

Run the set and display lines in the first terminal, then display the variable in the second. If it is absent there, that demonstrates the sessions have separate process environments for this variable. Clean up in the session where you set it. This test does not change a user-wide or machine-wide setting; it is a temporary example for the current PowerShell process. [S1]

A child process inherits environment variables from its parent. So, if PowerShell session A has the disposable variable set and starts a child process, that child receives the environment passed from A. [S1] That is different from a separate terminal session: the separate session does not become a child of A just because both are open on your screen.

The inheritance described here is from parent to child. The source does not specify how every application or shell displays variables, nor does it establish behavior for shells or operating systems beyond the stated PowerShell and Windows scope. [S1]

For a session-only change, set the value in the second terminal too, or start the program that needs it from the session where you set it. If you intend a broader Windows setting rather than a temporary process value, distinguish process scope from user and machine scopes: Windows treats those scopes separately. [S1] This guide does not establish the exact steps for changing persistent settings, so check the documentation for your PowerShell and Windows version before doing that.

For the disposable comparison, the expected pattern is: set in A, see it in A, check B, then remove it from A. A difference is evidence that you were comparing distinct session environments—not, by itself, proof of a problem with the application or terminal.