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We present results from a near-infrared (NIR) adaptive optics (AO) survey of pre-main-sequence stars in the Lupus Molecular Cloud with VLT/NACO to identify (sub)stellar companions down to $\sim$20 au separation and investigate the effects of multiplicity on circumstellar disc properties. We observe for the first time in the NIR with AO a total of 47 targets and complement our observations with archival data for another 58 objects previously observed with the same instrument. All 105 targets have millimetre ALMA data available, which provide constraints on disc masses and sizes. We identify a total of 13 multiple systems, including 11 doubles and 2 triples. In agreement with previous studies, we find that the most massive (M$_{\rm dust}$ $>$ 50 M$_{\oplus}$) and largest ($R_{\rm dust}>$ 70 au) discs are only seen around stars lacking visual companions (with separations of 20-4800 au) and that primaries tend to host more massive discs than secondaries. However, as recently shown in a very similar study of $>$200 PMS stars in the Ophiuchus Molecular Cloud, the distribution of disc masses and sizes are similar for single and multiple systems for M$_{\rm dust} < 50$ M$_{\oplus}$ and radii $R_{\rm dust}<$ 70 au. Such discs correspond to $\sim $80-90\% of the sample. This result can be seen in the combined sample of Lupus and Ophiuchus objects, which now includes more than 300 targets with ALMA imaging and NIR AO data, and implies that stellar companions with separations $>$20 au mostly affect discs in the upper 10$\%$ of the disc mass and size distributions.