Sort of makes sense to me. From what I understood from the wiki [1], you can compare the temperature regulation (very) roughly to a thermostat: There is a "target" temperature that the body "wants" to reach and a "current" temperature that the body senses it currently has.
If "current" is higher than "target", the body invokes several mechanisms to bring the temperature down, such as turning up sweat production and making you feel hot; if "current" is lower than "target", it does the same to raise the temperature e.g. by shivering and making you feel cold.
(Fever also works by changing the target temperature of that regulation system, which is why you feel cold at the beginning when the target was raised to the fever temperature and hot at the end when it was lowered back to the normal body temperature)
So it makes sense to me that shivering doesn't make you feel warm, because both, the feeling and the shivering are efforts by the body to raise the temperature.
I think there’s something else mentioned on that page that is a major contributor to feeling cold: redirecting blood flow away from the skin and in to the muscles (/organs)
If "current" is higher than "target", the body invokes several mechanisms to bring the temperature down, such as turning up sweat production and making you feel hot; if "current" is lower than "target", it does the same to raise the temperature e.g. by shivering and making you feel cold.
(Fever also works by changing the target temperature of that regulation system, which is why you feel cold at the beginning when the target was raised to the fever temperature and hot at the end when it was lowered back to the normal body temperature)
So it makes sense to me that shivering doesn't make you feel warm, because both, the feeling and the shivering are efforts by the body to raise the temperature.
[1] https://en.m.wikipedia.org/wiki/Thermoregulation_in_humans