Is a Dutch citizen living in Belgium traveling to Italy allowed to travel on an EU low cost company with his Belgian residency card?
Within the EEA+Switzerland you are allowed to travel, and enter, on a passport or a national ID card issued by one of the countries within the EEA/Switzerland.
National ID cards are issued by most of the EEA countries for residents (citizens and non-citizens), and contain information about your citizenship as well as the normal data. However, only citizens of an EEA country or Switzerland may use these cards as travel documents
An EEA citizen of country A, resident in country B and in possession of a national ID card from country B may use it for travel.
These cards may also be used for entering some non-EEA countries, such as Albania and North Cyprus.
A residency card is not a national ID card and is thus not applicable as a travel document. It is however up to the personnel at the check-in to accept or reject your travel document (when traveling within the EU), so in worst case scenario you can try to travel with the residence card. I have myself successfully done that a few times. Keep in mind though that even if you manage to fly out, you might have trouble going back, and I do not know if you're breaking any laws doing this.
As I understand it, under EU rules you should have a passport or ID card (if applicable) issued by your own country. There is a list of what counts as ID for each country. However, border officials should give EU citizens the opportunity to establish their nationality by other means.
Within the Schengen area, laws differ. Some Schengen members require all residents to have ID papers on their territory, and also to carry them during border crossings and to present them on demand to the relevant officials.
Some low-cost airlines go beyond the requirements of the law when checking papers, presumably to simplify their procedures and to minimize the risk of being fined for carrying illegals.
My wife's situation is somewhat analogous: she's an Irish citizen with a Belgian ID card. However, her ID card looks similar but is not the same as my (Belgian nationality) ID card. Hers is only usable within Belgium, for everyday identification purposes (e.g. stopped by police, or going to town hall).
We were explicitly told that it did not apply to travel, and that she would need to use her official passport for that.