What did my professor mean with this: gain-bandwidth product
The point of characterizing the gain-bandwidth product is that you will find that no matter what gain you configure using reasonable feedback resistors you will find

$$\left| A\right|\cdot f_{-3dB} \approx GBW$$

where \$GBW\$ is a characteristic of a particular op-amp.


  Now as I understand we have that the gain-bandwidth product is calculated multiplying the open-loop gain with the bandwidth.


As said above, it doesn't matter what gain configuration you use, you will get pretty close to the same gain-bandwidth product.

Using the open-loop gain might be convenient. But using a gain of 10 or gain of -2 config might be more convenient. 

Datasheets will normally say what gain they use, and it's not always open-loop, since loading effects and non-linearities will change the \$GBW\$ slightly.


  Now my professor tells me that the open-loop gain must be in decibels in that formula and therefore it should be about 23 kHz. I don't get why the open-loop gain needs to be in decibels in this formula, can someone please explain me?


You'll have to say exactly what formula you mean.

In the formula I gave above, \$A\$ is the linear gain, not the decibel gain. (So you'd use 100,000 rather than 100, if you were using the open-loop gain of an op-amp with a gain of 100,000 or 100 dB)