Dumb question: Why is gain and output on chart shown as minus db (-db)?

Just guessing but your hearing aids will be set to a certain level and there will be some adjustment range around that overall volume control range. It may just indicating where the normal level is compared to maximum.

A larger minus dB number (in an amplifier) is a lower sound level. In the output level chart, the smaller (positive) dB numbers are at the bottom and the larger (positive) dB numbers are a the top. The gain toggles in the middle probably hit zero as their top end and go negative from there as you toggle them down.

Typically, amplifier systems have a certain gain going straight through without any volume control on the input. Whatever that gain is, it’s called 0dB on the volume control. Volume controls in analog electronics are potentiometers (variable resistors) which divide the input voltage level, so you can only go down from maximum. The labels on the potentiometers start at 0dB at maximum and go down from there.

Of course, you can express the gain of your amplifier as so many dB and label the top end of the volume control at that number of dB. Say your amplifier doubled the sound level (voltage level). That would be 6dB of gain. Your volume control could be labeled 6dB at the maximum and go down from there. Where the amplification was neutral the label would be 0dB in that case, and when it became lower level than the input it would have negative dB numbers on the label. But the convention on volume controls is to put 0dB at the maximum and have all the numbers be in negative dB.

On a boost graph like in the software you pictured, they may be showing the gain of the amplifier on the curves.

Notably, and IMO strangely, the audiograms we make for our hearing profiles show positive dB for more hearing loss. Apparently their scale shows dB of loss rather than negative dB for “negative gain”. If I were inventing the audiogram I would have labled it as negative dB going down, but it’s the same thing, just looking at it from the opposite direction.