gdrt's answer provides an excellent summary of what occurred in the incident you provided as an example.
In general, a player cannot receive multiple "red cards" in the same match. The reasoning behind this becomes apparent when you refer to the sanction as a sending-off, rather than a "red card". The card is only a signal of the sanction.
Law 12 - Fouls and Misconduct, Section 3 - Disciplinary Action:
The yellow card communicates a caution and the red card communicates a sending-off.
It is obvious that a player cannot be sent-off more than once in a match. The Laws do not explicitly state this, but it is implied in Laws 3 and 12.
Law 12 - Fouls and Misconduct, Section 3 - Disciplinary Action:
A player, substitute or substituted player who has been sent off must leave the vicinity of the field of play and the technical area.
...
Only a player, substitute or substituted player may be shown the red or yellow card.
In Law 3 - The Players, Section 7 - Extra Persons on The Field of Play, the following language is used:
... a team official, substitute, substituted or sent off player ...
In Law 12 - Fouls and Misconduct, Section 4 - Restart of Play After Fouls and Misconduct, the following language is used:
... opposing player, substitute, substituted or sent off player, or team official ...
The distinction is made, as:
- when a player, substitute or substituted player is sent off, they become a sent off player
- only players, substitutes and substituted players can be sent off
- therefore, players who are already sent off cannot be sent off again - the red card or yellow card is not shown for subsequent sending-off or cautionable offences committed by a player after they have been sent off.
As happened in this case, if a player commits further misconduct (e.g. cautionable or sending-off offences) after having been sent-off, it is simply noted in the referee's report and the competition's disciplinary administrator or committee makes a determination on the matter.