Cannot use PSE 2024 on laptop when I go out?

The issue that Adobe Support may have had in mind is that you can't use the same Organizer catalog with two different versions of the program.  So, as Jeff says, you can have Elements 2025 installed on two different computers.  If you use the Organizer, it would be necessary to install your media and the catalog folder on an external drive which would be shared between the two computers.  If this is what you want to do, let us know if you need more details for that workflow.

Why did Adobe say you can't have both Photoshop Elements 2024 and 2025?

The license for Photoshop Elements 2024 is separate from 2025.

Many users including myself have multiple versions of Photoshop Elements on more than one computer.

Yes you can install Photoshop Elements 2025 on both computers and since Photoshop Elements 2025 doesn't need a serial number you could in theory install PSE 2025 on more that two computers and just sign in and out when required to keep the activated versions at two.

The only restriction Adobe has, as far as i know, is that a particular version, for example, Photoshop Elements 2024 can be installed and activated on two computers, but the user can't use both at the same time.