[n4r9]

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.

[telmo]

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

[the-smug-one]

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

[jolmg]

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.

[nachexnachex]

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

[the_af]

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