From my experience, when you register a trademark you generally have to specify it's use. For instance the Oilers can use 'Battle of Alberta' on t-shirts, stickers, or anything else that they specifically have the trademark for. The Esks, Rush, Oil Kings, Flames etc. could theoretically trademark it on items different than what the Oilers specifically trademark it on, but would usually have to give permission during the trademark process. If CIPO or USPTO (Canadian and US Trademark Offices) thought it was too similar on any of it's uses when it was applied for they would most likely disallow it wholly or on specific uses.
For the Rush, using the term in general or promoting it as an event it should be fine unless it's trademarked as an event of course. My guess is that the NHL would have the Winter Classic trademarked to prevent other companies from trying to make money on it, but it might be harder to trademark an somewhat regular occurrence such as Edmonton/Calgary hockey games.
Any term or phrase can have a TM behind it, but that usually means the phrase is being used by a company but not registered. When you apply to CIPO or USPTO any it get's approved, it gets you legal protection and you can use the R in a circle to tell users or imitators that it's registered and can face legal action.