Hi,
if the schema matches both of the databases, then you should be able to just create second DataContext instance (giving it the second connection string as an argument). The LINQ to SQL doesn't check in any way whether you use "the right" database - if it has the right columns & tables it will work.
However, LINQ doesn't automatically work with multiple databases in any "smart" way, so it will need to download the content to the memory before doing any operations that involve multiple data sources. You can still use single LINQ query to do this - but you have to be careful about what part of it is running using in memory data. (By the way, you can use extension methods like "ToList" to explicitly say - get the data from the databse at this point).
You also mention that the tables are nearly identical except for an ID in some case - does that mean that primary/foreign keys are different? In that case, some autogenerated relations may not work. If it means that there is a different column name, then you could manually edit the generated schema to contain both columns and then use only the right one. However, this feels a bit odd - unless you're planning doing some manual edits to the schema, you could as well just generate two very similar schemas.