No.
See Technical's post #2
The issue is the pin layout.
The serin/out and power +, 0 need to be in the same relationship to each other.
Manual1 has the pin layouts for the range of PICAXE chips.
You can see by comparing those diagrams that legs 1 & 2 and the pair opposite are always the same for 08, 08M, 14 and 20 so these could all fit in the same socket by ensuring Chip leg 1 is always in the socket leg 1 and avoiding pin damage (use a socket of the right size on top of the socket in the board).
For the others - , these relationships are different from each other, and from the 08/08M/14/20 group. The 18, 28 & 40 also need the reset circuit (or at least the pullup) adding.
So that is a minimum of four different incompatible layouts.
To do what you want you need a 'universal programmer'. Build it from strip board as suggested by Hippy, a breadboard, make a special, or incorporate the download circuit on your project. If you check the minimum operating circuits for each PICAXE you will find that virtually all the download components (except the connector) are required anyhow.
An example of a Universal programmer handling 8 thru 40 pin PICAXEs is shown below. The chip to program is selected by using the plug at bottom right and inserting it into the appropriate socket below the labels. This ensures all the necessary programming circuits are properly connected and includes power supply and reset.
This was built on a KPIC prototyping board and uses a ZIF (zero Insertion Force) socket to ensure no leg damage.