Joining the attributes of two nearly congruent streetnetworks (lines) in QGIS

QGIS 3.8 offers a new tool for this: Join attributes by nearest. You can find it in the toolbox. (For older versions theres the NNJoin plugin, which basically does the same, as csk already mentioned)

As joining lines can be complex (intersections, nodes and stuff like that, resulting in "incorrect" results), I suggest to go a workaround using points on lines (one per segment or line; or more, depending on your data) and perform the join on these. Once you joined the points, you can join the points to its line.