This property is rare in groups of small order. If I have calculated correctly then, of the $1237$ group of order at most $120$, $56$ have this property. But for $44$ of these, there is a unique group of that order, so there are really only $12$ interesting examples, which include $A_4$, $A_5$, ${\rm SL}(2,5)$, but not $S_5$, which has the same subgroup order type as $A_5 \times C_2$.

The principal reason that the property is rare is that a large majority of groups appear to satisfy the converse of Lagrange's Theorem: they have subgroups of all orders dividing the group order.


It would be very nice if there were statements of that kind that could be proved, but at present the available techniques do not seem to be adequate for this purpose.

Here is a sequence of reasonable conjectures of increasing strength:
<bl>
<li> Almost all finite groups satisfy the converse of Lagrange's Theorem.
<li> Almost all finite groups are supersolvable.
<li> Almost all finite groups are nilpotent.
<li> Almost all finite groups are $2$-groups.
<li> Almost all finite groups are $2$-groups of nilpotency class $2$ - you can add extra conditions to this.
</bl>

There are asymptotic estimates on number of groups with various properties
(see for example https://mathoverflow.net/questions/151491), but they are not precise enough to prove any of these "almost all" statements.

For your final question, the answer is of course no. It is never possible to distinguish between isomorphism types of groups using numerical data of this type. So you are really just asking someone to do a search for the smallest  counterexample, which is fortunately not too hard in this case: there are two pairs of groups of order $16$. One of the two pairs is $C_4 \times C_4$ and $\langle x,y \mid x^4=y^4=1, x^y=x^{-1} \rangle$, and the other pair is $C_2 \times C_8$ and $\langle x,y \mid x^8=y^2=1, x^y=x^5 \rangle$.