Basically could be 3 situations:
1) The state of the bike. The bike might seem too old or damaged to cycle with
2) Luggage. You might have received a big bouquet of flowers, someone might feel it's too big to carry by bike.
3) You might have an injury. When I broke a bone in my foot I went to hospital by bike (real story). "Ik heb een gebroken middenvoetsbeentje - Fiets jij ermee???"
You would not use "ermee" to refer to people (or teddy bears if you consider them company, not luggage). For people you would say "met hem/haar/hen".
I don't understand why "do you bike with this" is wrong. I understand this sentence like that : I have a very old bicycle - really old and rosty - and I pretend to bike from Paris to Amsterdam with it. And people telling me pointing at my old bike :Fiets jij ermee ? (do you bike with THIS ?"
How can you bike with it? You may bike somewhere, but you don't bike with it. With what? A cat, a dog, a headache" Plus how can you fiets ermee? The only thing you can bike with 'it' is the bike itself, or am I off my rocker?
If I remember correctly, mee is the form that met takes when it is compounded with another word. Note the separable verb meegaan.