For the past three years, Ramadan coincided with Tishrei, which a lot of people took note of. Tishrei is arguably the most important month of the Jewish calendar. Lots of holidays fall during it, including Rosh Hashanah, the most well-known Jewish fast day Yom Kippur (a 25-hour sunset-to-dusk fast), and another minor (dawn to dusk) fast day. Ramadan, of course, involves a daily fast (dawn to dusk) throughout the month. I know of some people who had joint Muslim-Jewish fast-breaking activities at the end of Yom Kippur in recent years, which is totally awesome.
But I think Elul is a better correspondence. Now, I know relatively little about Muslim observance in general. But from what I know, Ramadan is a month of relatively constant elevation of holiness and relatively consistant practices. This coincides with how Elul is often observed, with the twice-daily recitation of the "Psalm for the Days of Awe" as it's called in my prayer book (psalm 27), and the daily blowing of the shofar. (Of course, Ramadan's importance in Islam definitely exceeds Elul's importance in Judaism. It's not a perfect comparison.) In both cases, asing for forgiveness from one's misdeeds in the past is a prevalent theme. The Elul practices are meant as a lead up to Rosh Hashanah, the holiday that begins the moment Elul ends, a very important holiday full of rejoicing and 100 shofar blasts and lots and lots of good food. Similarly Eid ul-Fitr, one of the two most important festivals in the Muslim calendar by my understanding, begins the moment Ramadan ends, and is greeted with lots of celebration and lots and lots of good food. Next month we'll be celebrating together!
For those of you observing Ramadan, I don't know whether it's appropriate to wish you a happy one, but I think "meaningful" can't fail to be appropriate, so have a meaningful Ramadan. Those of you who are in tune to the Jewish calendar, chodesh tov, happy new month. And for everyone else out there, happy September! (And, Americans, happy Labor Day!)