I've seen reports like this, but they are all anectodal. In addition, I remember seeing similar articles in the past, for example around the time of the start of Schengen, so without numbers it's hard to know what to make of it.

I do have some figures for Israel, though. The Forward, quoting a survey (which I couldn't find) by the Menahim Begin Heritage Center says

A recent survey by the Jerusalem-based Menachem Begin Heritage Center found that 59% of Israelis had approached or intended to approach a foreign embassy to ask for citizenship and a passport.

With U.S. in slump, dual citizenship in EU countries attracts Americans
By Andrew Abramson, Palm Beach Post

For millions of Europeans who braved the Atlantic Ocean for a glimpse of the Statue of Liberty and dreams of a lavish life, there was little thought of ever emigrating back.

Yet for a new generation of Americans of European descent, the Old Country is becoming a new country full of promise and opportunity.

The creation of the European Union and its thriving economy is very appealing for Americans in a global economy.

"With an EU passport, I can live and work in 27 countries," said Suzanne Mulvehill of Lake Worth. "With a U.S. passport, I can live and work in one."

Americans can claim citizenship in any of the 27 European countries that are in the EU based on the nationality of their parents, or in some cases, grandparents and great-grandparents. Citizenship in one of those countries allows you to live and work in any EU nation.

Since the United States doesn't keep statistics on dual citizens, it's impossible to know exactly how many people have applied for citizenship in Europe. But it's estimated that more than 40 million Americans are eligible for dual citizenship, and a growing number of Americans want to try their luck elsewhere...

And yes, if I could I would too. It only makes sense to have multiple passports.