Design should without a doubt consider the target audience. It should be one of the main factors in your whole process.
I beileve that good design is universal
To an extent, yes—but design needs to be fit for purpose. What is good design for one situation isn't necessarily good for another.
Take for example the branding for an event. Is this a high end corporate conference or a local charity event aimed at families and children? Most likely the design for either would be unsuitable for the other.
For your specific situation it's hard to tell without seeing anything but it may not necessarily be bad design or unsuitable, it may just not be what the client had in mind and was expecting. If you can rationalize your design, the process, function and outcome then do that and convince the client. Otherwise you may just have to chalk it up to miscommunication and work closer to what your client expects.
It's worth keeping in mind that some clients will know exactly what they want, and that will sometimes be something very bad—it's your job to explain to your client why it is bad and rationilze a good solution.