Learn French: Je bois mon café tout en lisant le journal. - I drink my coffee while reading the newspaper.

Both mean 'while reading', but adding tout emphasizes the simultaneity of the actions. It strongly highlights that the coffee drinking and the newspaper reading are happening at the exact same time.

The lesson mentions tout en can show concession or simultaneity. Which one is happening here?

This is pure simultaneity (two things happening at once). If it were concession (meaning 'even though' or 'while still'), the context would contrast the two actions, such as Il n'a rien appris tout en lisant le journal (He learned nothing even while reading the newspaper).

Could I use this structure if I am drinking coffee while someone else is reading the newspaper?

No. The French gerundive (en + present participle) strictly requires both actions to share the same subject. Since Je is the subject of bois, Je must also be the one lisant (reading). If someone else were reading, you would need a conjunction like pendant que (e.g., pendant qu'il lit).

How exactly do we form the present participle lisant from the verb lire?

You take the present tense nous form (nous lisons), drop the -ons ending to get the stem (lis-), and add -ant. This is a great rule to remember because it applies to almost all French verbs.