Is my laptop equipped with a BIOS or UEFI?

In Windows 10, I go to System Information and see that my BIOS Mode is UEFI.

You have UEFI firmware.

I boot into an LM live disc and type sudo efibootmgr, which displays my boot entries.

Yeah, you have UEFI firmware.

When I contacted ASUS, they told me that if I don't have a setting in BIOS to switch between Legacy and UEFI, I don't have UEFI. I have a relatively new laptop that is only two years old.

That's not guaranteed. (It might be more true for ASUS specifically, if their tech support says so, but it's not mandatory in general.)

Most PCs did have such an option, although it doesn't always look the same, e.g. it is sometimes called "Legacy boot support" or "Compatibility Support Module". But not all PCs do – some firmwares always work in hybrid mode, allowing you to select from both UEFI and BIOS boot options at any time.

(Note that the option doesn't actually switch the whole firmware between BIOS and UEFI – it only activates or deactivates BIOS emulation so that you could boot disks in the BIOS way, while the rest of the firmware remains the same.)

Now at least Intel has repeatedly announced that they will no longer support BIOS mode (not even the BIOS compatibility mode in UEFI), so there already exist new PCs actually don't have this option anymore – they support only native UEFI-mode boot.

I've seen that UEFI has a graphical user interface, allows mouse use, and frequently has more features, whereas BIOS is more like old-school blue-ey settings

No, not really.

Having a GUI is merely common among new UEFI-based motherboards but by no means necessary. I think it's primarily Gigabyte UEFI on desktops that always looks fancy, but plenty of older UEFI firmwares have old-school blue-ey settings (e.g. those made by AMI or Aptio).

Meanwhile, plenty of BIOS systems had graphical mouse-based interfaces even ~25 years ago (mostly laptops from the Windows 95 era) – just take a look at this 486DX PC from 1994 which has an entirely graphical AMI BIOS setup screen.

In general, UEFI-based motherboards are more likely to have GUIs because they're new (UEFI makes some things easier, and larger flash capacity also makes some things easier).