An interesting Romance counter-example is "Obrigado" vs "Obrigada" ("thank you", in Portuguese). I'm not even remotely fluent, but if I recall correctly, the gender is dependent on the speaker, not the person being addressed.

I've heard of this example before, and this is speculative but I wonder if it's because the speaker is counter-intuitively also the object of the phrase.

The root "obrigad" sounds a lot like the English word "obligate". If this is more than just a coincidence, then a more direct translation is "I am obliged" rather than "thank you". Said this way it makes a lot of sense why the verb is determined by the speaker.

Yes, this is exactly the reason. (I am a native speaker)

Can it be extended with a reference to the (self-)object? Like "me llamo" being "I call myself" in Spanish for "my name is".

The difference between the "obrigado" in Portuguese example and the "me llamo" in Spanish example is that "obrigado" is probably an adjective while "llamo" is a verb. An equivalent Spanish example would be "estoy obligado" ("I am obligated" with male speaker) and "estoy obligada" ("I am obligated" with female speaker).

As an aside, I don't think Spanish has a word that's quite like "obliged". Maybe "endeudado", but that's just "in debt" and is not particularly about favors.

We can say "muy agradecido" or agradecida in formal context. It's unusual, but works as an example

In Spanish, "me llamo" carries no gender information.

Well, practically all adjectives in Spanish/Portuguese/French/Italian change depending on the grammatical gender of the noun they refer to. And, of course, if the noun in question is "I" (as in your case), then they reveal the (physical) gender of the speaker because usually

    grammatical gender of a person == physical gender of that person

But compared to the differences in, e.g., Japanese that some people here are discussing, this seems rather minor.

The most obvious English case would be "beautiful/handsome".

I am handsome.

I am beautiful.

Those words are differentiated on the object, not the gender of the speaker.

I was trying to give an example similar to obrigado/obrigada in English.

OK, but neither are examples of language being different based on the speaker, they are both based on gender of the object, whether is is contextually yourself, or another.

Handsome being a male adjective is a recent development though. Read some 19th century novels and the women are routinely described as handsome.

I told my girlfriend she was handsome the other day and she objected strenuously.

https://english.stackexchange.com/questions/355908/why-is-ha...

Oh, yes, but those aren't nouns. I didn't mean Spanish isn't gendered -- it obviously is -- but that gendered nouns aren't indicative of the gender of the speaker, so that particular observation was unrelated to the topic under discussion. Your example is indeed an example of gendered language depending on the speaker, and has similar examples in Spanish (e.g. "contento" / "contenta" for "happy").

Seeing that "obrigato" is used to say thanks kind of makes me wonder if Japanese's "arigato" has a similar etymology.

Apparently not:

http://www.accessj.com/2014/05/etymology-of-arigato.html

There are other little things in Portuguese:

I did it myself: Eu fiz eu mesmo/Eu fiz eu mesma.

And most adjectives. Actually, "obrigado" is "obliged" so the phrase "much obliged" is equivalent to the Portuguese "muito obrigado".

"Obrigado" is a contraction for "I am" obrigado. ("I'm indebted to you"), so you need to use the gender appropriate for the speaker