Measure part length of polygon using QGIS

As indicated in my comment this is the workflow that I would follow:

1. First buffer your sea by a small amount, the value of small depends on the spatial reference units you're working in and how well the polygon data follows your coast. Try to use the smallest number you can.
2. If you don't already have a unique polygon ID then add a unique polygon ID field.

3. Degenerate your polygons into lines, the generated lines will have all the attributes of the contributing polygon (which includes the unique ID).

4. Clip the polygon lines with the sea buffer to extract the part that is adjacent to the sea only.

5. Calculate the lengths of lines into the attribute table, there may be multiple sections from a the same source polygon within the clip but that's ok for now.
6. Generate statistics from the clipped lines using the unique ID as a classification field and length as a statistic field, this will give you the SUM of all the lengths adjacent to the sea for each polygon unique ID. This table can then be joined back to the original polygons if that is your requirement.

I would use the same process in Esri or any other GIS software, all that would change is the links to the instructions.