Is it true that same characteristic and minimal polynomial ⇒ similar?

No. 4×4 nilpotent matrices with Jordan blocks of sizes {2, 2} and {2, 1, 1} Both have χ = x⁴ and m = x², but different numbers of blocks (ranks 2 vs 1).

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Is this true?

Same characteristic and minimal polynomial ⇒ similar

No — it is false.

The counterexample

4×4 nilpotent matrices with Jordan blocks of sizes {2, 2} and {2, 1, 1}

Both have $χ = x^{4}$ and $m = x^{2}$ , but different numbers of blocks (ranks 2 vs 1).

The kind of mistake this is

Invariants don't determine

Two objects sharing an invariant were treated as the same object.

Drill statements like this

linear algebra canonical forms

Others that fail the same way

Same characteristic polynomial ⇒ similar
Diagonalisable ⇒ invertible
Real matrix with real eigenvalues is diagonalisable
AB = I ⇒ BA = I for all matrices
Every real symmetric matrix is positive definite if det > 0
AB and BA have the same minimal polynomial

From Eigenvalues and Canonical Forms › Eigenvalues, characteristic & minimal polynomials, Cayley–Hamilton

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