No, each EU country has its own electoral system. The EU is currently a federation of mostly independent countries. Most or all of these countries have voter ID laws. This is usually not a problem since everyone can be reasonably assumed to have a government-issued ID document.

To give an example, in Germany your birth is always documented. As a citizen you always receive an ID card when you turn 16; in fact, you are legally required to have a valid one that includes current information. For example, if you move you have to make an appointment with the citizens' center in your new municipality to change your registration and they'll put a correction sticker with your new address on the ID card. If you change your name you have to get a new one, which costs about 50 bucks; same if you lose yours or if it expires. People under 24 and poor people get rebates, in the latter case of up to 100%.

We don't have a voter roll per se; all citizens receive voting invitations by mail several weeks in advance of an election. These are used in conjunction with your ID card to get access to your assigned polling place. The law mandates a certain density of polling places so nobody has to travel too far or wait for too long. The invitation also includes everything needed to register for mail voting including a postage-paid return envelope so poverty isn't a factor in whether you can vote by mail.

So a lot of the criticism about voter ID laws you hear in the United States doesn't apply here because ID cards are ubiquitous (and have been for a long time) and we go out of our way to make sure poor people aren't disadvantaged.