In light of all that we know these days about disease causes and agents, the word “germs” is a terrible relic that means … practically nothing. It’s a catch-all term that covers “every infectious agent”, and in that way it has been rendered useless.
The word covers bacteria, viruses, parasites and parasitic protozoans, even worms, and so many and varied types of each, sometimes even including the various vectors that they use to enter our bodies and infect us, that it just doesn’t belong in a modern vocabulary.
As far as we know – because our knowledge always changes, so I’m not going to say that “this is absolutely true” – what we know as flu is caused by various kinds of fairly rapidly-mutating viruses. This is why “the flu vaccine” is concocted anew each winter season in northern latitudes, in an attempt to meet the expected threat to come later in “the flu season” (winter). I do not know if hydrogen peroxide is an effective virucide, but even if it were, “dipping the thermometer into the bottle” is a very ineffective way to sterilize the instrument, as you have realized. Doing that effectively contaminates the entire bottle, unless the contents really are an effective virucide – which I doubt in the first place. (And even if the peroxide is a good virucide it’s still poor practice to risk contaminating one’s entire supply of the cleaning agent by dipping the soiled or used instrument into it.)
So I would not recommend the use of the contents of that bottle any more for medicinal or other internal purposes, but as far as laundry, general cleaning (except wound cleaning) and other uses it should be perfectly fine. I would also shy away from recommending it for ear cleaning. Even though there are no mucous membranes that would be at direct risk of attack by a stray surviving flu virus (since the risk is lessened by the dispersal of the small amount of virus likely to have been on the instrument into the larger quantity of liquid peroxide), there are indirect pathways between the ear, nasal passages and throat which could become susceptible to a virus entry through the ear canal.