Is it considered bad style to assign values to variables like this?

x = "foobar" or None
y = some_variable or None

In the above example, x gets the value ‘foobar’.

No, it’s a common practice. It’s only considered bad style for expressions that are considerably longer than yours.

The primary danger of doing something like this is the possibility that (in the second case) some_variable is False but not None (the integer 0, for instance) and you don’t want to end up with y equal to None in that case.

I also feel a bit unconfortable using that kind of expressions. In Learning Python 4ed it is called a “somewhat unusual behavior”.
Later Mark Lutz says:

…it turns out to be a fairly common coding paradigm in Python: to
select a nonempty object from among a fixed-size set, simply string
them together in an or expression. In simpler form, this is also
commonly used to designate a default…

In fact, they produce concise one-line expressions that help to eliminate line noise from the code.
This behavior is the basis for a form of the if/else ternary operator:

A = Y if X else Z

OP’s syntax is perfectly fine.
The official name for “assignment with or” is null coalescing and there’s actually a Wikipedia page about it now! https://en.wikipedia.org/wiki/Null_coalescing_operator
This question may be useful as well:
Is there a Python equivalent of the C# null-coalescing operator?