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My nation (Spain) consists of the mainland (north of the swamp tile, contains Madrid) and 3 or so continents (Las Palmas is on one). I had the idea of connecting them to make unit transport easier. I am also building freights on the Palmas continent to set up routes with my ally, whose on the Madrid continent. Because trade gets a bonus when two cities are on different continents, I am worried the bridge might make Freeciv consider them to be one continent. Are the continents still two different continents?

I believe they are no longer different continents. You should be able to middle-click (there might be some combination of shift-ctrl-alt-click that works as well, but it escapes me at the moment) to display various tile information; somewhere among that should be a continent ID. You could check if that is different between the two cities.

Yes, continents are recalculated any time a tile terrain is changed between land and ocean by any means. If there is a path, the continents are same.

You can know by middle-click the continent number of any tile you have ever seen, even if you have not seen it for long. Negative numbers are oceans/lakes, positive are land masses.

I have confirmed that Freeciv did consider the two continents to be a single continent. I assume a diagonal connection between two landmasses would still make them one continent. I then tested trade to see if it gets recalculated after separation. As demonstrated, it did boost trade. This seems like a mechanic I could use to boost trade with any nation, by creating little islands on which trade cities sit. and watch extra money flow because it's a "continent."