Nintendo Switch does not run Linux, it runs a proprietary OS called Horizon based on the Nintendo 3DS firmware. Not sure but it might or might not have some BSD code in the network stack or something.
Pretty much everything uses BSD source code in the network stack, including Windows, so that much is a safe assumption, but there's far more that the Switch is using.
According to the copyright notice, it uses the FreeBSD kernel. This tracks with reported use of BSD jails, which are part of the BSD kernel.