At the Pesach Seder the Urchatz is the washing of hands without a bracha. At the Pesach seder our table is like the altar, so since the Kohanim should wash their hands and feet in order to serve or make offerings, is the Urchatz symbolically referring to this washing? Or does it refers to another temple ritual? I wonder how it was established and by which basis we keep the urchatz ritual at the Pesach meal. Why don't we do Urchatz at for example the weekly Shabbat meal?
Good try, but no, not that washing.
In the late Second Temple period, it became common practice for most Jews, even non-kohanim, to try to keep all food as non-tamei as possible, as if it were terumah or a sacrifice. (Terumah -- not just bread, but even wine or oil -- would require hand-washing before consumption.) Thus, they would wash before any vegetable-dipped-in-liquid.
Over the next few centuries, we wound up in a situation where everyone is tamei anyhow, but the practice of washing stayed around for the staple -- bread -- while it was relaxed for all other foods. However on Seder night, we are reenacting what it would be like in Temple times, so we wash before eating vegetables-dipped-in-liquid.
(A carrot growing in the ground can't become tamei because it's attached to the ground. Pull it out of the ground, still can't become tamei hence no washing required. It can only become tamei once you rinse it.)