The short answer is ‘NO’. I know that, but anyway…
But what if I clear a raid with my guild, can I the next day join a group of my friends and clear the same raid with the same difficulty again? I know that it is possible in LFR, but you can get your loot only with the first kill on the week. Is the same possible on Normal and Heroic difficulties? Or if I want to raid with my guild I have to level a second character?
In modern non-Mythic raids, you do them more than once, but only get loot from the first kill. (Just like LFR.) Just make sure that you aren’t the group leader for the second raid, as the raid leader defines what bosses are still left to kill.
If the raid is part of Warlords of Draenor, Legion, BfA, or Shadowlands you can run the normal or heroic raids as much as you like; as Udiza said you’ll only get loot the first time you kill a boss each week for each difficulty level. So if you run one of those raids once on normal and once on heroic, you’ll get loot from each boss both times.
If the raid is from Wrath of the Lich King, Cataclysm or Mists of Pandaria then it gets more complicated as you can only kill each raid boss once per lockout period. So for example, if your regular raid group manages to kill the first 6 bosses in Icecrown Citadel on normal mode, you can’t go on to participate in a heroic Icecrown Citadel raid later that same week unless they’ve already killed those same 6 bosses. (The one exception to this is the Siege of Orgrimmar raid from Mists which follows the boss loot rule used by the newer raids.)
If the raid is from Vanilla or Burning Crusade, those raids only had one difficulty level so you can only run them once per lockout. In addition you are instance locked so you can’t join a different raid group even if they’ve already killed the same bosses that week.