Not an answer, just a small (but maybe influential) data point, which njuffa probably also looked up:

Euler in his "Elements of Algebra" ~1765 seems somewhat inconsistent:

Before introducing the notation $√$ he declares that square roots can be positive or negative:

125. What we have said in the preceding chapter amounts to this: that the square root of a given number is that number whose square is equal to the given number; and that we may put before those roots either the positive or the negative sign.

He then introduces the $√$ operation without being explicit as to whether it is multivalued:

130. […] a particular sign has been agreed on to express the square roots of all numbers that are not perfect squares ; which sign is written thus $√$ and is read square root. Thus, $√ 12$ represents the square root of $12$, or the number which, multiplied by itself, produces $12$ ; and $√ 2$ represents the square root of $2$ ; […] and, in general, $√ a$ represents the square root of the number $a$.

Somewhat later, it seems like he considers $√ a$ to be only positive:

131. When the number before which we have placed the radical sign $√$, is itself a square, its root is expressed in the usual way; thus, $√ 4$ is the same as $2$ ; $√ 9$ is the same as $3$; $√ 36$ the same as $6$.

But a bit later he treats it as multivalued:

150. We have before observed, that the square root of any number has always two values, one positive and the other negative; that $√4$, for example, is both $+2$ and $-2$, and that, in general, we may take $√ a$ as well as $-√ a$ for the square root of $a$.

According to Cajori, the radical sign $√$ was introduced 1525 in Die Coss by Christoff Rudolff, but I have not looked into that.

Not an answer, just a small (but maybe influential) data point, which njuffa probably also looked up:

Euler in his "Elements of Algebra" ~1765 seems somewhat inconsistent:

Before introducing the notation $√$ he declares that square roots can be positive or negative:

125. What we have said in the preceding chapter amounts to this: that the square root of a given number is that number whose square is equal to the given number; and that we may put before those roots either the positive or the negative sign.

He then introduces the $√$ operation without being explicit as to whether it is multivalued:

130. […] a particular sign has been agreed on to express the square roots of all numbers that are not perfect squares ; which sign is written thus $√$ and is read square root. Thus, $√ 12$ represents the square root of $12$, or the number which, multiplied by itself, produces $12$ ; and $√ 2$ represents the square root of $2$ ; […] and, in general, $√ a$ represents the square root of the number $a$.

Somewhat later, it seems like he considers $√ a$ to be only positive:

131. When the number before which we have placed the radical sign $√$, is itself a square, its root is expressed in the usual way; thus, $√ 4$ is the same as $2$ ; $√ 9$ is the same as $3$; $√ 36$ the same as $6$.

But a bit later he treats it as multivalued:

150. We have before observed, that the square root of any number has always two values, one positive and the other negative; that $√4$, for example, is both $+2$ and $-2$, and that, in general, we may take $√ a$ as well as $-√ a$ for the square root of $a$.

According to Cajori, the radical sign $√$ was first introduced 1525 in Die Coss by Christoff Rudolff like this "In this algorithm the square root, for reasons of brevity, will be denoted with the character √ , so √4 is the square root of 4." but Rudolff's book does not seem to consider negative numbers, so the question of multivaluedness is obsolete.