That said, Abb isn't another name for G - it's a distinct thing which, when played on typical musical instruments, is played the same way as a G. Basically note names to note frequencies is a many-to-one mapping.
If you want to understand the hairy details start by googling just vs equal temperament. But I went down that rabbit hole, and it hasn't made my procedural music any better, so I wouldn't really recommend it.
- A pitch standard, like A440, which anchors one arbitrarily chosen note to a specific frequency.
- A tuning system, which defines the frequency ratios between notes.
Almost all modern music assumes _A440_ and _12-tone equal temperament_. In 12TET, adjacent flats and sharps map to the same frequency. It's said that they are _enharmonic_. But there are many other systems where this isn't true. In _just intonation_, notes are defined by simple frequency ratios, and there is no enharmonicity at all. Every number of sharps and flats map to different frequencies.
For string or wind instruments playing without keyboards, there is no temperament. Each note represents a range of possible pitches, and what pitch you choose depends on the notes around it. For example, if you are repeating or holding a note, you don't want it to change pitch. But if the harmony around that note is changing, it may be in tune at one point in time, then out of tune at a later one. String quartet playing involves a great deal of working out compromises about what pitch each note should have.