If 冬 in itself means winter, do adding 天 Impact the meaning, or perhaps refer to winter time instead of the season itself? Or is it because 冬 is monosyllabic and 冬天 has two syllables?
In short, for disambiguation, at least oral disambiguation.
Most of the words in Old Chinese are monosyllabic words. However, when it comes to oral speech, it is too ambiguous to use. As time went by, we developed Modern Chinese, which encourages the usage of disyllabic words, for oral disambiguation.
In oral speech, the monosyllabic word "dōng" can mean 冬 (winter), 东 (east)， 鸫 (thrush), 氡 (radon), etc, so it is too ambiguous to pronounce "dōng" only. Therefore, in Modern Chinese, we use the disyllabic word "dōng tiān", written as 冬天, literally "winter days", for disambiguation. You can also use the disyllabic word "dōng jì", written as 冬季, literally "winter season", for oral disambiguation.
Similarly, there is a monosyllabic word 锋 (fēng) and a monosyllabic word 利 (lì) in Old Chinese. Both of them means "sharp". However, in oral speech, "fēng" can mean 锋 (sharp), 风 (wind), 疯 (mad, crazy), etc, and "lì" can mean 利 (sharp, benefit), 力 (force), 粒 (grain), etc, so it is too ambiguous to pronounce either "fēng" or "lì". Therefore, in Modern Chinese, we use the disyllabic word 锋利 (fēng lì) for "sharp", for oral disambiguation.
Can I just write monosyllabic words but speak disyllabic words? No. They look too archaic. Nowadays, native Chinese speakers are familiar with Modern Chinese, but unfamiliar with Old Chinese. If you write monosyllabic words, native Chinese speakers may feel like "have you just time-travelled from thousands of years ago?" Actually, it is almost similar to the feeling if you speak Classical Latin in Europe.
You can use: 冬、冬季、冬天 even 萧条期 can mean winter: a bleak, desolate period. Why not have more than 1 word with the same meaning? That makes writing more interesting.
Winter is a word whose origin is unknown. It may be related to an old word for white, but that is just a guess.
In English you may say winter, or wintertime. As Drooze mentioned, maybe 冬 surfaces more, but not solely, as an adjective.
We might say: a winter coat
but I would not say: a wintertime coat
Robert Louis Stevenson called winter Winter-Time, nowadays we write wintertime.
Which brings me to his poem:
Winter-Time Robert Louis Stevenson - 1850-1894
Late lies the wintry sun a-bed,
A frosty, fiery sleepy-head;
Blinks but an hour or two; and then,
A blood-red orange, sets again.
Before the stars have left the skies,
At morning in the dark I rise;
And shivering in my nakedness,
By the cold candle, bathe and dress.
Close by the jolly fire I sit
To warm my frozen bones a bit;
Or with a reindeer-sled, explore
The colder countries round the door.
When to go out, my nurse doth wrap
Me in my comforter and cap;
The cold wind burns my face, and blows
Its frosty pepper up my nose.
Black are my steps on silver sod;
Thick blows my frosty breath abroad;
And tree and house, and hill and lake,
Are frosted like a wedding-cake.
Now, I'd like to tell you about the origins of 天, and my theory about aliens in Ancient China, but I'll save that for another time .....