It should depend on the UPS, some I believe are indistinguishable from wall power, especially without actually trying.
Even for the cheaper UPSes I wonder if the issue isn't the cycle rate, but the cycle shape? My understanding is that they tend to be able to hit 60Hz pretty easily, but the cheaper ones are a ~square wave instead of a ~sine wave. Maybe the digital clock just glitches on that more.
To clarify this comment: there are two primary types of UPS, double conversion and line interactive.
Double conversion takes power from AC, converts it to DC to charge batteries, takes battery output, and inverts it back to AC. All power drawn from the UPS goes through the battery and inverter stack, and there is no transient/power loss when AC mains are lost. They tend to be more expensive, louder, run hotter, etc.
Line interactive UPSs, on the other hand, tend to be cheaper and are in most cheap consumer products. They take AC mains, convert it to DC, and charge batteries. But AC mains is also connected directly to the output device through switch circuitry that will quickly switch the power source from AC mains to batteries/inverter if power loss is detected.
Reputable UPSs will use pure sine wave inverters for converting DC battery back to AC. Modified sine waves are indeed a lot cheaper but are not suitable for some sensitive equipment.
Even for the cheaper UPSes I wonder if the issue isn't the cycle rate, but the cycle shape? My understanding is that they tend to be able to hit 60Hz pretty easily, but the cheaper ones are a ~square wave instead of a ~sine wave. Maybe the digital clock just glitches on that more.