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We present constraints on the massive star and ionized gas properties for a sample of 62 star-forming galaxies at $z\sim2.3$. Using BPASS stellar population models, we fit the rest-UV spectra of galaxies in our sample to estimate age and stellar metallicity which, in turn, determine the ionizing spectrum. In addition to the median properties of well-defined subsets of our sample, we derive the ages and stellar metallicities for 30 high-SNR individual galaxies -- the largest sample of individual galaxies at high redshift with such measurements. Most galaxies in this high-SNR subsample have stellar metallicities of $0.001<Z_*<0.004$. We then use Cloudy+BPASS photoionization models to match observed rest-optical line ratios and infer nebular properties. Our high-SNR subsample is characterized by a median ionization parameter and oxygen abundance, respectively, of $\log(U)_{\textrm{med}}=-2.98\pm0.25$ and $12+\log(\textrm{O/H})_{\textrm{med}}=8.48\pm0.11$. Accordingly, we find that all galaxies in our sample show evidence for $α$-enhancement. In addition, based on inferred $\log(U)$ and $12+\log(\textrm{O/H})$ values, we find that the local relationship between ionization parameter and metallicity applies at $z\sim2$. Finally, we find that the high-redshift galaxies most offset from the local excitation sequence in the BPT diagram are the most $α$-enhanced. This trend suggests that $α$-enhancement resulting in a harder ionizing spectrum at fixed oxygen abundance is a significant driver of the high-redshift galaxy offset on the BPT diagram relative to local systems. The ubiquity of $α$-enhancement among $z\sim2.3$ star-forming galaxies indicates important differences between high-redshift and local galaxies that must be accounted for in order to derive physical properties at high redshift.