Depending on your trade in the Navy, you could be posted to different types of units (Ships are technically considered a Unit)
Most trades that are shipboard specific, such as Bos'n, or W Eng Tech, are usually posted to a ship after they have been initially qualified. There are some instances where those trades are able to be posted ashore, such as instructor positions, or other job openings that require someone of that rank at a certain location (such as an instructor at BMQ), and there are a few Navy-specific trades that I know of that can be posted to different locations. For example, a steward is a Navy-specific trade, but I was just talking to a Master Seaman who works in the accomodations cell at CFB Borden.
Also, if your trade is a "purple" trade, such as RMS Clerk, you can be posted anywhere, regardless of element. I know of a few RMS Clerks working in CFBs across the country, and they all wear the Navy uniform.
As for the identifiers on the uniform, I know that when you are posted to a ship, you will have a ball cap with the ship's name and Registration number on it, as well as the ship's crest gets sewn on your NCD jacket, above your name tag. As for how other units do it, I can only speak for CFB Halifax and CFLRS (the recruit school in St. Jean sur Richelieu). I know that CFLRS issues crests to their Naval personnel, and there are other units within CFB Halifax that issue crests and ball hats to their personnel (such as the Naval Engineering School, MARLANT, etc.) I'm almost sure it'll be the same way in Esquimalt, but as for other units or other CFBs, I can't tell you, because I just don't know.
Hope this helps.
Rev