Is it safe to download the Logitech M310 Unifying Receiver program directly from Logitech's homepage???

One shouldn't need any software to use a Logitech Unifying Receiver. These just plug into a Mac USB port and talk to the compatible Logitech device via radio. More software, and especially any software that may be incompatible with the current operating system, can lead to unplanned Mac behavior.

Have you confirmed on the Logitech site that the M310 device is compatible with Macs, and in particular, the current operating system?

I use a logitech mouse ms anywhere 2s

it's safe unless you judge logitech to be untrustworthy or unable to keep their mac software free of malware.