Step 1: Understanding the Concept:
Statement I deals with the cyclic structure of carbohydrates. Statement II compares the configurations of the lower carbons in aldohexoses and ketohexoses.
Step 2: Detailed Explanation:
Statement I: Anomers are cyclic diastereomers that differ in configuration only at the hemiacetal/hemiketal carbon ($C_1$ for glucose). $\alpha$ and $\beta$ glucose are indeed anomers. Thus, Statement I is True.
Statement II: In the open chain structures of D-glucose and D-fructose, the chiral centers at $C_3, C_4$, and $C_5$ have identical configurations (OH on right, H on left for $C_4/C_5$ and OH on left for $C_3$ in Fischer projection). This is why they yield the same osazone. Thus, Statement II is True.
Step 3: Final Answer:
Both Statement I and Statement II are true.