Looking at a Japanese dictionary, there's nothing about it being a counter in there. Here's the definition for 円 http://www.weblio.jp/content/%E5%86%86
Contrast that with the definition for 本, which does list it as a counter (助数詞) http://www.weblio.jp/content/%E6%9C%AC
That's an interesting question. I haven't seen it listed as a 助数詞 in any Japanese dictionaries, though I think I have seen one or two lists of counters featuring it. I think this is a question of how you define what a counter word is. How strict your definition needs to be depends on the context on which you're talking about counters. Let me talk about two potential definitions.
A counter is simply "word used to count things". In this case, yeah, 円 is a counter, but so is any other unit, like 米/メート. Here the word 'unit' though, gives us a conceptual framework to differentiate them from counters. Counters are really for classes of words/objects, while things like 円 and 米 are units of a specific measurement (in one case, monetary value measured in yen, the other, distance measured in meters). This definition is still useful for learners and general discussion about Japanese, though, since it highlights the unified way of actually counting in Japanese. Whether the word fits in the following, stricter, definition or not, you would still use it to count things in the same way,
A counter is a word that must be attached to a number to represent the class of objects being counted. So a counter is a grammatical construction that classifies another word and is required to make it countable. This definition is much more interesting in a grammatical perspective, for discussing how Japanese works in detail or comparing with other languages. If you think about it, there aren't really countable words in Japanese without counters. There's a lot to be said about the differences between these and measure words, but I think that's beyond the scope of this post. One note is that counters are a part of grammar. If you think about it, using 円 is not really the same as using 羽 to count birds or 本 to count long objects. A counter does not express its meaning without a number or 何 associated with it. That isn't to say it can't have other meanings when alone (like say, 台), or feature the same/related meaning, but with another pronunciation (like 人). You can, however, use 円 alone to refer to the currency itself, or you can discuss the conversion rate between 米 and 里. So counters, in a way, both require and are required by a number to accomplish the desired count.
Having all that said, since this is a dictionary, it's probably better to keep to a technical definition, particularly when it comes to word classification. In the case of JMDICT in particular, definitions and classifications are usually sourced from Japanese dictionaries, where 円 is not classified as a counter.
@Leebo Thanks for your reply! I know it isn't listed as a counter in dictionaries, I just observed that it appears to be used as a counter so I was wondering why it wasn't classified as such.
@Zengor Thank you for your detailed reply! Your explanation made everything a lot clearer. I think you're right - 円 is a unit like メートル, not a counter, so I guess it is a type of noun. I still think it would be great if it was classified as "noun, unit" or something, so students can differentiate them from normal nouns, but I understand now that it's not quite the same as a counter. I assumed it was like a counter because that was how I explained it to students who wanted to know why they didn't need to put が after it, but after your explanation I realise you're right. Thank you so much!
Isn't 円 a counter as well as a noun?
We say 1000円ある, not 1000円がある (most of the time), the same as 1枚ある instead of 1枚がある. To me, this suggests that 円 is a counter and should be classified as such in the dictionary. Does anyone have any other opinions?