Both styles are named after the dojo where they originated. The 館 ("kan") in both names means a type of building and is commonly suffixed to the names of dojos.
Looking at the name of the Shodokan dojo, the 道 ("do") is the same "do" used in the names of various Japanese arts including aikido itself. It does indeed mean "way" as in "the way of X". The 昭 ("sho") is a little more complicated - the founder of the style, Kenji Tomiki, cited it as being derived from the name of the Showa era (1926-1989, the reign of Emperor Hirohito) during which the dojo was founded.
The kanji of Showa are 昭 ("sho", which has meanings such as "shining", "glory" and "enlightenment") and 和 ("wa", which generally means either "peace" or "Japan"). The name of the Showa era is officially derived from a quote from classical Chinese literature, 百姓昭明 協和万邦, which means something along the lines of "If all men are open about their intentions, the world can achieve peace". In his speech at the founding of the dojo, Kenji Tomiki referenced this quote as reflecting the spirit of cooperation that the dojo represents.
The Shotokan dojo, on the other hand, is essentially simply named in honour of its founder - 松涛 ("Shoto") is a pen-name that Funakoshi Gichin had been using since his youth. If that pen-name itself has any significant meaning, it's less clear. The characters themselves literally mean "pine waves", and there's a district of Shibuya in Tokyo that has this name, so it's probably just derived from the placename one way or another.