How does partially thawing a frozen solution affect concentration?
In general, no, a frozen solution that partially thaws is not the same concentration as the original solution.

The reason for this is the same reason chemists can purify a compound by crystallizing it, and one of the reasons why north polar ice cap is freshwater, despite floating in salt water.

When something (e.g. water) crystallizes (e.g. forms ice) the crystal lattice tends to reject impurities (e.g. things dissolved in the water). This means that as the solution crystallizes/freezes, the solid portion becomes depleted in the solute, and the remaining liquid portion becomes enriched in the solute. Eventually, though, if things are cold enough, even the concentrated residual liquid will freeze. But this mean that you now no longer have a uniform concentration of solvent in your frozen sample.

The reverse happens when you thaw it. Since you don't have a uniform concentration of solute, not all portions of your solid sample have the same freezing point depression. So what tends to happen is that portions of the solid which are more concentrated in solute will thaw first (as they have a lower melting point), whereas portions which are depleted in solute will thaw later. -- This is all generalization, though. What actually thaws first depends a lot on the details of where these regions are in relation to the heat source, and how thermal energy is flowing through the sample.

The up shot of this is, though, that if you freeze a stock solution, you can't partially thaw it and expect to get the appropriate concentration. You have to completely unthaw the stock solution and mix it thoroughly to make sure you have a homogenous mixture of the correct concentration.

This is why it is typically recommended to make single-use aliquots of stock solutions like antibiotics. Not only do the smaller aliquots thaw faster, but you don't put your entire stock solution into multiple freeze/thaw cycles, which can degrade some of the more delicate compounds.