There is a clear, self-contained classification of compact, connected Lie groups in "Lie Groups: An Approach through Invariants and Representations" by Claudio Procesi. See Chapter 10, Section 7.2, Theorem 4, page 380. It says such a group is of the form $K_1\times\dots\times K_n\times T/Z$ where each $K_i$ is connected, compact, and simply connected, $T=(S^1)^m$ is a compact torus, and $Z$ is a finite subgroup of the center. Furthermore two such groups $K_1\times\dots\times T/Z$ and $K'_1\times\dots\times T'/Z'$ are isomorphic if and only if there is an isomorphism $K_1\times\dots\times T\rightarrow K'_1\times\dots\times T'$ taking $Z$ to $Z'$. Note that the numerator is specified by a list of types $A_n,B_n,\dots, E_8$ and the integer $m$. The result is obtained from a bijection between compact connected and reductive algebraic groups, and is equivalent to the classification by root data.

It isn't entirely elementary to tell when two such groups are isomorphic. For example there are precisely $3$ compact groups of rank 2 and semisimple rank 1:

$SU(2)\times S^1$ ($Z=1$)

$SO(3)\times S^1$ ($Z=\langle -I,1\rangle$)

$U(2)=SU(2)\times S^1/\langle (-I,-1)\rangle$

There are lots of other finite subgroups of $\mathbb Z_2\times S^1$, but they all give one of these three groups.

For another example the following two groups are not isomorphic, where $\zeta=e^{2\pi i/5}$:

$SU(5)\times S^1/\langle(\zeta I, \zeta)\rangle \simeq U(5)$

$SU(5)\times S^1/\langle(\zeta I,\zeta^2)\rangle$

There is a clear, self-contained classification of compact, connected Lie groups in "Lie Groups: An Approach through Invariants and Representations" by Claudio Procesi. See Chapter 10, Section 7.2, Theorem 4, page 380. It says such a group is of the form $K_1\times\dots\times K_n\times T/Z$ where each $K_i$ is connected, compact, and simply connected, $T=(S^1)^m$ is a compact torus, and $Z$ is a finite subgroup of the center satisfying $Z\cap T=1$. Furthermore two such groups $K_1\times\dots\times T/Z$ and $K'_1\times\dots\times T'/Z'$ are isomorphic if and only if there is an isomorphism $K_1\times\dots\times T\rightarrow K'_1\times\dots\times T'$ taking $Z$ to $Z'$. Note that the numerator is specified by a list of types $A_n,B_n,\dots, E_8$ and the integer $m$. The result is obtained from a bijection between compact connected and reductive algebraic groups, and is equivalent to the classification by root data.

For example there are precisely $3$ compact groups of rank 2 and semisimple rank 1:

$SU(2)\times S^1$ ($Z=1$)

$SO(3)\times S^1$ ($Z=\langle -I,1\rangle$)

$U(2)=SU(2)\times S^1/\langle (-I,-1)\rangle$.

For another example the following two groups are not isomorphic, where $\zeta=e^{2\pi i/5}$:

$SU(5)\times S^1/\langle(\zeta I, \zeta)\rangle \simeq U(5)$

$SU(5)\times S^1/\langle(\zeta I,\zeta^2)\rangle$