Interestingly, the way we dream (or at least how we remember them) appears to have been heavily influenced by the type of media we consumed as children:

The Black-and-White Era: Up until the 1960s, studies often found that the majority of people reported dreaming in black and white. One 1942 study of college students found that only 29% reported occasional color in their dreams.
The Shift: As color television and films became ubiquitous, reports of colorful dreams skyrocketed. Today, research from The Conversation and The Washington Post shows that younger generations (who grew up with HD color screens) almost exclusively dream in color.
Persistent Greyscale: About 12% of the general population still reports dreaming only in black and white, a phenomenon most common among those over 55 who were exposed to black-and-white media during their critical developmental years.

Why We Might Forget Color
Even if you dream in color, you might not always remember it. Experts from Harvard Medical School suggest a few reasons for this:

Attention and Salience: Just like in waking life, if the color of someone's shirt isn't important to the "plot" of your day, you won't commit it to memory.
Emotional Intensity: Vibrant or unusual colors (like a neon-pink banana) are more likely to be remembered because they grab our attention or hold specific emotional significance.
Muted Palettes: Many people report that their dreams aren't strictly black and white but feature soft, "washed out" or pastel tones rather than vivid hues.

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I think you and I started with black and white TVs, however early on we did get into color TVs. With this being said, my dreams appear to be colorized however, I also agree with the comment above about the color of something normally is not relevant and not stored in memory. With most my memories through they are mostly colorful and I can remember small details such as the way a Christmas tree was decorated, the color of some clothes and other items as well as the general colors of the scenes themselves.

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