Avconv option -c:a copy

Yes, it might actually make sense if you are being forced to change formats.

If you have a file with 95kbps in a highly efficient format, to retain the same quality, a relatively inefficient format as mp3 needs a higher bitrate.

Of course you will never get anything back that was lost in the first place. On the contrary, encoding as mp3 will reduce the quality further. Every lossy format uses other means to reduce the amount of data that is stored, by (simplified) throwing away "unneeded" parts of the data. Round trip through a bunch of different formats and there won't be much left ...

So if you want to stay as close a possible to the quality your file has now, you should chose a higher bitrate. 320kbps are probably wasted space, but for mp3 something in the order between 128 and 192 is needed to maintain - or at least come close to - the quality of a more efficient 95kbps file.

It works fine, except for the quality because ffmpeg is using a very low bitrate.

The defaults for mpeg4 are not well chosen, so the target bitrate is quite low. Specify your own target with -b:v 2M (depending on your resolution) or even better, use constant quality with -q:v 5 (as suggested by Mulvya in the comments). In the latter case, lower values mean better quality.

My question now is if I need to specify a quality/bitrate or if it is possible to tell ffmpeg to use the original bitrate somehow.

No – it wouldn't even make sense to do that. If you are going from one codec to another, they might be less or more efficient. Each codec offers different quality at different bitrates – for example, H.264 is much more efficient than MPEG-4 Part II. H.265 is 30–50% more efficient than H.264, etc. The same goes for the actual encoders implementing the codecs: x264 is more efficient than the reference H.264 encoder, etc.

Therefore, taking the original bitrate may not work well. And even keeping the same codec, if you re-compress, you might want to use an even higher bitrate to avoid generation loss.