Unfortunately, I am not aware of big test data sets for real world low-rank matrices. However, I can point you to other low-rank matrices, which might be interesting for testing the performance of your algorithm.
Low-rank matrices from sampling functions
You can generate low-rank / numerically low-rank matrices by sampling bivariate functions on a grid, i.e. $M_{ij} = f(x_i,y_j)$ for some sets of points $\{x_1,\dots,x_n\}$,$\{y_1,\dots,y_n\}$.
A list of potentially interesting functions is contained in the testbattery of Chebfun2 (https://github.com/chebfun/chebfun/blob/master/tests/chebfun2/test_battery.m). They also have a test battery for trivariate functions, should you also be interested in low-rank approximations of tensors of order 3.
Such matrices/tensors do not contain real world data, but their low-rank approximations are still crucial for algorithms in numerical analysis. One big difference to real world data matrices is, however, that you can evaluate the function related matrices with almost no noise.
Remark. Note that you can generalize the concept of low-rank matrices to low-rank functions. A function is of rank $r$ if it can be written as $f(x,y) = \sum_{i=1}^r g_i(x) h_i(y)$ for some univariate functions $g_1,\dots,g_r$,$h_1,\dots,h_r$. If you sample form a rank-$r$ function the resulting matrix has at most rank-$r$.
Sampling supposedly real-world like matrices
If you are interested in generating matrices that supposedly behave like real world data matrices, you might want to have a look at the paper "Why Are Big Data Matrices Approximately Low Rank?" (
https://epubs.siam.org/doi/pdf/10.1137/18M1183480). The authors model real world data matrices using a special stochastic model, for which the generated matrices tend to be numerically low-rank.
Note, however, that the assumptions in that paper might not match with the data matrices you get in certain applications.
