It might come easily to you, but me and other people I know often draft emails and such 3-4 times, asking for proofreading from trusted friends.
Yea, the proofreading for docs and writing client copies isn't fun, but the skills to do these well help all of your written communication immensely.
Your org could be very different, but I feel like I am recognized in large part because I care about communicating, and put a lot of effort into clear emails and docs. Part of your recognition could be because you have great writing skills.
The benefits don't accrue as much when you're writing material you've been told to write, especially when it's not something the recipient will be happy to receive. Those can be valuable learning experiences, but it's not harvest season yet.
The value comes when you CHOOSE to compose messages meant to persuade YOUR peers and internal stake holders. Your skill with the pen gives you a voice, like a bard who can urge a crowd to action.
What words do we associate with those kinds of people?
Influence
Leadership
Writing can help get you there
It's been recognized by my peers and managers. Not much has ever really come from it. I've been given opportunities to draft "bad news" copy to be sent to clients, and I'm the point person for editing design documentation. But none of these are really roles of prestige. If anything, I'm really seen as being the person responsible for the grunt work that people don't generally enjoy doing.