Here is a string of comments which might be helpful.

**UPDATE** at the end I conjecture an upper bound $a(n) \leq \lfloor (\frac{n-1}{2})^2 \rfloor$ which satisfies a stronger property.

Consider instead cases of $$\prod_1^k(x_i+a)= \prod_1^k(y_i+a) \tag{*}$$ where the multisets $\{x_1,\cdots ,x_k\}$ and $\{y_1,\cdots ,y_k\}$ are disjoint. I'll assume the elements are listed in increasing order. To stick to the OP, add the requirement that the $y_i$ are distinct. For example, $a(5)\geq 2$ because there are counter-examples to $a=0$ and $a=1.$
$$(2+0)(2+0)(3+0)(2+0)(5+0)=(1+0)(2+0)(3+0)(4+0)(5+0)$$
$$(2+1)(2+1)(3+1)(3+1)(4+1)=(1+1)(2+1)(3+1)(4+1)(5+1)$$ Cancel out common factors to to see that sources of these counter-examples are $1\cdot 4=2 \cdot 2 $ and $2 \cdot 6=3 \cdot 4.$ In the other direction, one can pad an example of $(*)$ by changing the right-hand side to $\prod_1^n(i+a)$ and adding on the left the same new factors. Here $n$ could be $\max(x_k,y_k)$ or anything larger.

- The final remark exhibits that $a(n)$ is non-decreasing.

Of the values reported so far the larger ones are somewhat close . $$a(14)=33 \lt 42=\lfloor (\frac{13}2)^2\rfloor$$

$$ a(15)=45 \lt 49$$
Here is a potential conjecture. It is false. I mention it only because the counter-examples are lovely.

Suppose that the value of $\prod_{i=1}^n (a + x_i) -\prod_{i=1}^n (a + y_i)$ is independent of $a$. Does that mean that the shared value is $0$ and $x_i=y_i?$

The answer is no because of ideal solutions to the Prouhet-Tarry-Escott problem. For example $2^k+3^k+7^k=1^k+5^k+6^k$ for $k=0,1,2.$ This explains the observation that $$(2+a)(3+a)(7+a)=42+41a+12a^2+a^3$$
$$(1+a)(5+a)(6+a)=30+41a+12a^2+a^3$$ so the two always differ by $12.$

The OP is to find the *first* $a$ which satisfies the condition.

for any set of integers $(x_1,...,x_n)$ and $1\leq x_i \leq n$:

$(x_1,\dotsc,x_n)$ is a permutation of $(1,\dotsc,n)$ if and only if:

$(x_1+a)\dotsb(x_n+a)=(1+a)\dotsb(n+a)$.

I will instead seek the *last* $a$ which fails the property. This (plus $1$) is then an upper-bound on $a(n).$

I will conjecture that for fixed $n,$ this last bad $a$ is at most $(\frac{n-1}{2})^2.$ My justification is sketchy and would probably benefit from classical inequalities.

By my comments above, given $n$, a particular $a$ is bad if there is $k$-member subset of $\{a+1,\cdots ,a+n\}$ and a disjoint multiset of $k$ elements from the same set which have the same product.

I think that the extreme case is $k=2$ with $a+1=s^2$ and $n=2s+1$ so $a+n=(s+1)^2$

Then $s^2\cdot (s+1)^2=(s^2+s)\cdot (s^2+s).$

Here are plots showing that $a=18$ and $a=45$ are good for $n=11$ at least as far as $k=2.$

The first shows that there are no solutions of $18\cdot 29=u \cdot v$ with $19 \leq u,v \leq 28$ The hyperbola $xy=19\cdot 28$ (on this interval) snakes through the lattice points without hitting any of them. That isn't surprising given that $19$ is prime.

The second shows the hyperbola $xy=45\cdot 56.$ Along the diagonal are the lattice points $(x,101-x).$ The diagonal below is the closest lattice points. But the hyperbola stays above that closest diagonal.Hence there are no solutions of $u \cdot v=2520$ in that range other than the endpoints.

The $a$ chosen for these is larger than needed but it makes the plot easier to see.

In the cases mentioned above such as $ 25\cdot 36=30 \cdot 30$ , the hyperbola is tangent to the lower diagonal and the contact point is a lattice point.

I'll suffice to end this sketch by saying, without justification, that for larger $k$ the surface $x_1x_2\cdots x_k=y_1y_2\cdots y_k$ lies below the hyperplane $x_1+x_2+\cdots +x_k=y_1+y_2+\cdots + y_k$ which is rich in lattice points. If the numbers are large enough then that surface stays close enough to the hyperplane that it never touches the parallel hyperplane of nearest lattice points. It seems as if “large enough” decreases with $k$. A study of the known bad a values might make that clear. Do any of the known counter examples use more than $k=2?$

The exact value of $a(n)$ in the OP depends on the distribution of fairly composite integers in certain intervals of length $n.$ That is not very predictable.However I think the simplifications here might make the searches easier. The values reported so far seem close to the bound.