This paper is probably relevant for your question: MR0528966 Wolpert, Scott The length spectra as moduli for compact Riemann surfaces. Ann. of Math. (2) 109 (1979), no. 2, 323–351.
And here is a more recent paper on the subject:
MR3770180 Parlier, Hugo Interrogating surface length spectra and quantifying isospectrality. (English summary) Math. Ann. 370 (2018), no. 3-4, 1759–1787.
These papers suggest that finitely many geodesic lengths never determine the surface.