You're right - from the simple math in the numbers you've given, you can deduce that each average neuron makes thousands of connections (sending and receiving).
Of course, this varies a bit by cell type:
There is one special synapse called the Calyx of
Held which is a nearly
one-to-one connection between certain types of cells in the auditory
brainstem: one globular bushy cell makes only 1 or 2 calyx synapses with MNTB
neurons (though they also have collaterals to other targets) which themselves
receive 1 or at most 2 calyx synapses (though they also receive a small
fraction of input from other sources) (Smith et al 1991, Smith et al 1998,
Rodríguez‐Contreras et al 2006)
Napper and Harvey 1988 estimate about 175,000 synapses arrive at each Purkinje
cell in the cerebellum.
The ballpark number of "thousands of synapses per neuron" is
reasonable for most cells, however.
Neurons can and do make multiple synaptic connections with the same post-synaptic cell - that's one of the ways to regulate the strength of connection between two neurons - but most synapses will be from different neurons.
Synapses on different compartments of a cell can have different effects, because they tend to sum more with other local connections and less from far ones, and because synapses closer to the cell body are more likely to directly control firing than synapses far away.
Axons from many cell types in neocortex have a complex branching pattern, where axons often branch both locally as well as sending various collaterals to other areas of neocortex and even to subcortical structures (see some examples in Ojima et al. 1991 & 1992). They make synapses all along these branches except for the longest projection stretches.
Hawkins, J., & Ahmad, S. (2016). Why neurons have thousands of synapses, a theory of sequence memory in neocortex. Frontiers in neural circuits, 10, 23.
Napper, R. M. A., & Harvey, R. J. (1988). Number of parallel fiber synapses on an individual Purkinje cell in the cerebellum of the rat. Journal of Comparative Neurology, 274(2), 168-177.
Ojima, H., Honda, C. N., & Jones, E. (1991). Patterns of axon collateralization of identified supragranular pyramidal neurons in the cat auditory cortex. Cerebral Cortex, 1(1), 80-94.
Ojima, H., Honda, C. N., & Jones, E. G. (1992). Characteristics of intracellularly injected infragranular pyramidal neurons in cat primary auditory cortex. Cerebral Cortex, 2(3), 197-216.
Rodríguez‐Contreras, A., de Lange, R. P., Lucassen, P. J., & Borst, J. G. G. (2006). Branching of calyceal afferents during postnatal development in the rat auditory brainstem. Journal of Comparative Neurology, 496(2), 214-228.
Smith, P. H., Joris, P. X., Carney, L. H., & Yin, T. C. (1991). Projections of physiologically characterized globular bushy cell axons from the cochlear nucleus of the cat. Journal of comparative neurology, 304(3), 387-407.
Smith, P. H., Joris, P. X., & Yin, T. C. (1998). Anatomy and physiology of principal cells of the medial nucleus of the trapezoid body (MNTB) of the cat. Journal of neurophysiology, 79(6), 3127-3142.