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Consider the following groups: $(\mathbb{Z}_4,+)$, $(U_5,.)$, $(U_8,.)$ and the set of symmetries for a rhombus if I am not mistaken the first and last are equivalent. What other justifiable equivalencies and nonequivalencies are there and what does it mean rigorously to be in the same group in general?

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Two groups $G$ and $H$ are "the same" if there is an isomorphism between them, ie a map $f: G\to H$ which is bijective ans such that for all $g,h\in G$ it satisfies $f(gh) = f(g)f(h)$ (so here the multiplication on the left is inside $G$ and on the right it is inside $H$). The reason that this is the condition we want is that in that case, anything we can say about $G$ which only uses that it is a group, we can transfer to $H$ via $f$ and vice versa.

The first two groups you mention are isomorphic, while the thirds is not isomorphic to the first two. What the symmetries of a rhombus are depends on whether you allow a square to be a rhombus. If you do not, then the group of symmetries of a rhombus is isomorphic to $U(8,\cdot)$

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The term you are looking for is called a group isomorphism.

By the way if your $(U_8,\cdot)$ denotes $(\Bbb{Z}/8\Bbb{Z})^{\ast}$, it is not isomorphic to $(\Bbb{Z}_4,+)$ because one is cyclic while the other is not.

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