Is it a 230V/460V transformer or a 230V/230V transformer?
It's for the radio's HV DC supply.
Full-wave rectifiers were not easily created using valve technology. The centre-tapped transformer allows for easy full-wave rectification using two valves (tubes) or two rectifiers combined into one tube.
Figure 1. A full-wave tube rectifier. Each anode (on the left) will conduct when its supply goes positive. Image source: Electronics Notes.
See the linked article for much more on the topic.
What is the purpose of this "center tap"?
That should be clear by now.
Is there an alternative to achieve the exact same thing with "modern parts"?
Yes. A single secondary and a semiconductor bridge rectifier.
Isn't this exact the same thing like using P1 and P2 as P3 and P4? If no, why?
No. One of the mains wires will be neutral and so will have zero volts with respect to earth. Doing as you suggest would result in half-wave rectification.
In addition the transformer provides isolation between the mains and the radio. This may offer some protection in the event of a high voltage fault or problem with neutral.