The wolfram-engine package is available from the official repository

https://archive.raspberrypi.org/debian/pool/main/w/wolfram-engine/wolfram-engine_13.3.1+20230814893_arm64.deb

and can be simply installed or updated using the built-in package manager (apt).

❯ sudo apt update
❯ sudo apt install wolfram-engine
Reading package lists... Done
Building dependency tree... Done
Reading state information... Done
wolfram-engine is already the newest version (13.3.1+20230814893).
0 upgraded, 0 newly installed, 0 to remove and 0 not upgraded

That said, I do get a install-wolfram-engine-13.3.1.sh file after clicking the manual installation link at https://wolfr.am/raspi-install, so perhaps there is some cache somewhere that needs to be cleared.

Thanks a lot for your quick and finally helpful response, Ilian Gachevsky! Install with apt did not work for me, I only got 13.0. Indeed I succeeded to download the install-wolfram-engine-13.3.1.sh file on my Windows 10 PC. But it did not install correctly on my Raspberry Pi. Now I suspected that the problem could be my old 32 bit Bullseye Raspberry OS. Indeed installing and running 13.3 worked like a charm on the new 64 bit Bookworm OS! Does 13.3 work only on 64 bits and/or Bookworm?