It isn't the same principle. Firstly, coastlines are more jagged because they are hit by waves perpendicularly, while rivers are shaped by water flowing along the banks, smoothing them. Secondly, the borders are typically defined by either the thalweg (greatest depth) or median line, either being smoother lines than the banks. Thirdly, river borders are then in practice defined by measuring the coordinates of particular sample points along the idealized line and then using straight lines or simple mathematical curves to connect these, forming a simple non-fractal boundary.
Exactly. The coastline paradox is a mathematical curiousity, not a practical objection to measuring things. Coastlines are not infinite length in practice. You define a system of measurement then a length in that system
It also makes sense to pick a resolution because coastline changes on an hourly basis (and minute-basis, for rivers during rainfall), so that these differences would not massively affect the measurement every single second