If you tried to use alias givens rather than given instances (with-syntax)
inline given maxForDoubles: Max[Double] = new Max[Double]:
  inline override def max(a: Double, b: Double): Double = if a < b then b else a
inline given maxForPairs[A, B](using mA: Max[A], mB: Max[B]): Max[(A, B)] = new Max[(A, B)]:
  inline override def max(x: (A, B), y: (A, B)): (A, B) = (mA.max(x._1, y._1), mB.max(x._2, y._2))
this wouldn't work because of
Implementation restriction: nested inline methods are not supported
But with with-syntax nested inline are allowed because
inline given maxForDoubles: Max[Double] with
  inline override def max(a: Double, b: Double): Double = if a < b then b else a
inline given maxForPairs[A, B](using mA: Max[A], mB: Max[B]): Max[(A, B)] with
  inline override def max(x: (A, B), y: (A, B)): (A, B) = (mA.max(x._1, y._1), mB.max(x._2, y._2))
is actually
class maxForDoubles extends Max[Double]:
  inline override def max(a: Double, b: Double): Double = if a < b then b else a
inline given maxForDoubles: maxForDoubles = new maxForDoubles
class maxForPairs[A, B](using mA: Max[A], mB: Max[B]) extends Max[(A, B)]:
  inline override def max(x: (A, B), y: (A, B)): (A, B) = (mA.max(x._1, y._1), mB.max(x._2, y._2))
inline given maxForPairs[A, B](using mA: Max[A], mB: Max[B]): maxForPairs[A, B] = new maxForPairs[A, B]
Indeed, with -Xprint:pickleQuotes option (or -Xprint:typer) the with-syntax produces
//    given class maxForDoubles() extends Object(), App.Max[Double] {
//      override inline def max(a: Double, b: Double): Double = 
//        (if a.<(b) then b else a):Double
//    }
//    final inline given def maxForDoubles: App.maxForDoubles = 
//      new App.maxForDoubles():App.maxForDoubles
//    given class maxForPairs[A >: Nothing <: Any, B >: Nothing <: Any](using 
//      mA: App.Max[A]
//    , mB: App.Max[B]) extends Object(), App.Max[
//      Tuple2[maxForPairs.this.A, maxForPairs.this.B]
//    ] {
//      A
//      B
//      protected given val mA: App.Max[A]
//      protected given val mB: App.Max[B]
//      override inline def max(x: Tuple2[maxForPairs.this.A, maxForPairs.this.B]
//        , 
//      y: Tuple2[maxForPairs.this.A, maxForPairs.this.B]): (A, B) = 
//        Tuple2.apply[A, B](this.App$maxForPairs$$inline$mA.max(x._1, y._1), 
//          this.App$maxForPairs$$inline$mB.max(x._2, y._2)
//        ):(A, B)
//      def App$maxForPairs$$inline$mA: App.Max[A] = maxForPairs.this.mA
//      def App$maxForPairs$$inline$mB: App.Max[B] = maxForPairs.this.mB
//    }
//    final inline given def maxForPairs[A >: Nothing <: Any, B >: Nothing <: Any]
//      (
//    using mA: App.Max[A], mB: App.Max[B]): App.maxForPairs[A, B] = 
//      new App.maxForPairs[A, B](using mA, mB)():App.maxForPairs[A, B]
Let's temporarily simplify the with-syntax removing using and maxForDoubles
class maxForPairs[A, B]/*(using mA: Max[A], mB: Max[B])*/ extends Max[(A, B)]:
  inline override def max(x: (A, B), y: (A, B)): (A, B) = ??? // (mA.max(x._1, y._1), mB.max(x._2, y._2))
// inline given maxForPairs[A, B]/*(using mA: Max[A], mB: Max[B])*/: maxForPairs[A, B] = new maxForPairs[A, B]
Then the difference between
val inst: maxForPairs[Double, Double] = new maxForPairs[Double, Double]
inst.max((10.0, 3.0), (20.0, -7.0)) // compiles
and
val inst: Max[(Double, Double)] = new maxForPairs[Double, Double]
inst.max((10.0, 3.0), (20.0, -7.0)) // doesn't compile: Deferred inline method max in trait Max cannot be invoked
is understandable because there is a rule
Inline methods can also be abstract. An abstract inline method can be implemented only by other inline methods. It cannot be invoked directly:
abstract class A:
  inline def f: Int
object B extends A:
  inline def f: Int = 22
B.f         // OK
val a: A = B
a.f         // error: cannot inline f in A.
 
https://docs.scala-lang.org/scala3/reference/metaprogramming/inline.html#rules-for-overriding
And max is exactly an abstract inline method in the trait Max.
summon[Max[(Double, Double)]] returns the precise type of implicit:  maxForPairs[Double, Double] <: Max[(Double, Double)] (as shapeless.the in Scala 2), not just Max[(Double, Double)] (as implicitly in Scala 2).
But if we restore using and maxForDoubles then for some reason this confuses inlines: both options for val inst (: maxForPairs[Double, Double] and : Max[(Double, Double)]) produce Deferred inline method max in trait Max cannot be invoked.
This seems to be a bug or underspecified feature. Try to open a ticket at https://github.com/lampepfl/dotty/issues
By the way, sometimes with-syntax behaves weirdly in comparison with alias givens (even without inlines):
https://github.com/lampepfl/dotty/issues/8882