I seem to lack (almost) completely any sense of rhythm and tempo. When I go to the dance floor I usually have troubles moving in sync with the music if the base is not really simple and clear, not because I'm unable to dance but because I'm unable to understand what's the basic structure of the songs being played. I'd like to know if there's a way, and what that would be, to learn and feel the music. I'd really love to develop this gift that seems to be innate in every person walking on this planet, except me.
There is a difference in feeling the rhythm and not responding to the rhythm with certain bodily parts(for example, when dancing). Some muscles(, muscle groups) are more trained to receive commands that others and this is really based on luck when not trained.
A way to feel the rhythm is trying at first to count by beating your foot or hand or even counting in your head. In particular, in popular dance music the rhythm is based on the 4/4 time signature, that means that there are 4 somehow accented beats and you can easily understand them. Just count , , ,  in the same spacing and repeat in a way that you feel right, it will probably be right, you can recognise that if on countng in the new  the song feels like it starts repeating with you, if that does not happens it means tha song uses a different time signature(different way to count) but you might as well recognise it instinctively at some point and start counting right.
By recognising that you can really count correctly shortly you could be able to feel the beat(the flow) with all of your body. Remember that dancing is art. You don't have to recognise rhythm to be able to dance, just dance and if you don't try too much you will start stepping on rhythm automatically.
By practicing you will gain control of the muscles you're practicing on and it will become easier.