This is a popular question!

And actually, they did have something similar to an ID card, in the form of wooden plaques or bronze diplomas.

Here is one answer I gave in this thread :

The same way they do today: official paperwork!

During the Republic era the tribal lists were overseen by the censors, and these were effective citizenship lists. Later in the Empire Marcus Aurelius reformed the rolls, so that in Rome the list was held at the treasury of Saturn. In the provinces the lists were likewise held by prefects of the treasury.

Now these were administrative lists, but individuals also had records. Normal citizens had to register the birth of a citizen child within thirty days and would receive a wooden plaque. These had a list of seven witnesses who vouched that the child was a citizen.

Citizenship could also be granted for military service. We have many bronze military diploma for soldiers granted citizenship.

We also know how these records were used. We have many letters - primarily from Egypt where the dry conditions preserved papyrus - concerning people accused of being runaway slaves. These people claimed that they were citizens, but they had to prove it. If they didn't have their diploma or wooden diptych they had to write to central or regional authorities who would check their records.

Sources: Historia Augusta Marcus Aurelius and Sherwin-White The Roman Citizenship

Edit: Added image of bronze military diploma

Also in that thread /u/searocksandtrees posted five more good threads on this topic, which you can see here .