For their more common meanings, like in your paragraph, as a verb you want affect, and as a noun, effect. So, when in doubt, use that as a rule of thumb.

However, both have alternative meanings as the other part of speech.

Affect as a noun means emotion or disposition, and is mostly used in psychology. Your psychologist may say you have a depressed affect.

Effect as a verb means to bring about. You might say that a successful protest effected change in society.

As a verb, in addition to “have an impact on,” affect can also mean “to pretend to have,” like “she affected an air of mystery,” although this is less common.

"Effect" and "affect" are hilariously messed-up. They have subtly overlapping definitions sometimes but other times mean totally different things. They look almost the same in writing. They can sound almost the same. In spoken English, for some senses of each word we denote what we mean by changing the sound ("affect" may be pronounced almost like "effect", or, for one of its noun definitions and a related verb definition, very differently) or stress (for "effect", in some cases we hit the second syllable a little harder than other times).

The way you used "effect" here, its verb sense of "to bring about or cause" is the one that suggests itself, which isn't what you meant.

The simple way to keep the words' overlapping meanings straight, is that it's "effect" when it's a noun, "affect" when it's a verb. "Effect" can also be a verb, and "affect" can be a noun, but those definitions don't overlap.

Your post did indeed call for "affect", as you suspected.

"Effect" as a verb means to bring about, or to bring it into existence. "Affect" means to have influence on them.

It's definitely wrong in that paragraph.

Companies will often state a subsidiary is wholly owned by the ultimate parent regardless of which tier the subsidiary is at. The Thunderbird subsidiary could be under the Firefox subsidiary and the statement would still be true.

I agree that it should be "affect". Affect doesn't look wrong to me:

  and technology choices made in Firefox can and do affect Thunderbird, just like they effect e.g. Zen Browser or Tor Browser.

I'm no expert on the rules of english, but I think maybe it would be slightly more gramatically correct to say that "choices made in Firefox can and do have an effect on Thunderbird". I would probably have phrased it like that. Maybe that's why it looks wrong to you?

English is a bit of a bastard language IIUC, and so we accept the way you've phrased it too, but in that case it should be "affect".

I hope this helps rather than making things more confusing! ;)