Step 1: Find the molecular formula of each compound.
Write both structures clearly. Methoxyethane is $CH_3-CH_2-O-CH_3$. Propan-1-ol is $CH_3-CH_2-CH_2-OH$.
Step 2: Count the atoms.
Both have 3 carbon atoms, 8 hydrogen atoms, and 1 oxygen atom. So both are $C_3H_8O$.
Step 3: Note the rule for isomers.
Isomers are compounds with the same molecular formula but different structures. Here both share $C_3H_8O$, so they are isomers.
Step 4: Look at the functional groups.
Methoxyethane has an ether link, the $-O-$ in the middle of the chain. Propan-1-ol has an alcohol group, the $-OH$ at the end.
Step 5: Match to the right name.
When two isomers have different functional groups, the type is called functional group isomerism.
Step 6: Conclusion.
An ether and an alcohol with the same formula are a classic pair of functional group isomers.
\[ \boxed{\text{Functional group isomerism (Option 3)}} \]