Nope; these sentences share a common root-level grammar production, that's all.

They are not identical. They are similar at this production step:

  S -> subj verb obj

The verb is common:

  verb -> ... -> bit

So we have the same overall sentence pattern with the same verb, in the same tense.

Other than that, the derivation diverges:

  subj -> ... -> a dog
  subj -> ... -> my friend

You can't just ignore substructure differences and symbol identities in comparing trees!

Of course, any two things are the same, if you choose to ignore anything that is different and define "sameness" that way.

Ignoring the symbols is exactly what is meant by "the grammar" of a sentence. Two things are grammatically the same if they use the same grammar productions when parsing/producing.

This isn't conveniently bending definitions to make myself sound correct. This is the specific technical definition that was worked out by linguists in their empirical examination of language (and later adopted by computer scientists because it happens to work nicely with programming languages too). It might not match with some colloquial uses, but it's a reasonable assumption that an article discussing linguistics would be using the precise linguistic definition.

Are they though? Isn't 'a' an indeterminate article and 'my' a pronoun?

I get the point of the example, but it seems different at the grammatical level too.

Subject/object should be at a different level above grammar.

It may be easier to understand the point being made here if we consider what it means to be entirely absent these grammar. AIUI, but can't come up with a concrete link on the Internet quickly, Koko, the gorilla that knew some ASL, tended to speak in sentences like "Koko koko cup cup cup koko koko want want want cup cup cup", if they were transcribed literally. With such utterances, there is no way to distinguish between "Koko wants the cup" and "The cup wants Koko", or even "Koko wants the cup to want Koko".

There are animals that have demonstrated some "semantics". There are parrots that do better than Koko with spoken language, but still do not produce anything like the grammatical constructs humans do. Everything I've seen out of parrots could be covered by memorizing a handful of grammar constructs in a very brute-force manner, and they have no deliberate ability to combine them.

The core point is that to distinguish '"A dog bit my friend" and "My friend bit a dog."' requires grammar, which so far as I know as one who tends to pay attention to these things, is not a thing that animals have ever demonstrated a deep understanding of (if there are animals that could distinguish that, that is probably their maximum limit) and certainly never demonstrated production of. (Or, at the very least, we've never decoded and proved is speech, if you're thinking of whales and dolphins and such.) If you produce and understand speech as the Koko sentence above, you can't distinguish those two sentences. It's not about whether changing subject and object makes a "different sentence" by some metric, it's about whether the animal in question has a well-defined usage of subject and object as distinct grammatical categories at all.

For example, in my sentence "The cup wants Koko" I used earlier, it is invalid, yet everyone here reading it has at the very least the same understanding of why it is invalid, because what it says and the grammar of why it says that is completely clear to us.

Syntactically, they're both determiners. Semantically, they're a bit different since possessives (like 'my') aren't closed under permutation†, while many other determiners ('some', 'every', 'most' &c.) are closed under permutation.

† An operation mapping a set X onto another X' by relating every member of X with some member of X', so that X' has the same number of members as X, but the identity of the members may be different.

Others have addressed most of this, but no one else has called this out specifically: "my" is not a pronoun. Pronouns stand in for nouns. "my" stands in for a determiner. Both replace concrete with relative, but that's a semantic category that doesn't impact syntactic class.

"A dog bit a friend" and "my dog bit my friend" still work the same, though.

Indeed, I think it's just an incorrect example, the point gets across anyway.

Substitute 'my' for the 'a's?

You can't be randomly substituting words for other words (let alone ones with different lexical categories) and keep claiming that you have the same parse tree!

Agreed, and of course the meaning also changes - What I mean is that you can have a different example that makes the point while avoiding this particular issue.