Golf the Chinese 9*9 multiple table

Stax, 66 characters

9mYF"得一二三四五六七八九"cacy*~@ny@\p;11AH:b!n;A/X@]z?px'十z?p,A%sn@]z?' +qD

Number of bytes depends on the encoding used for Chinese characters.

Run and debug online!

Explanation

9mYF...D    Loop `n` times and print a newline after each loop, `n`=1..9

"..."cay*~@ny@\p
"..."c              Push the string and duplicate it
      ay            Fetch the outer loop variable
        *           Multiply with the inner loop variable
         ~          Move the product to input stack for later use
          @         Take character at the index specified by inner loop variable
           ny@      Take character at the index specified by outer loop variable
              \p    Print the two characters

;11AH:b!n;A/X@]z?p
;11AH:b!        ?p    Is the product not in range [11,20)?
                      Output (*) if true, (**) if false.
        n;A/X@        Character at the index of the "ten" digit of product
              ]       Convert character to string (*)
               z      Empty string (**)

x'十z?p,A%sn@]z?' +q
x'十z?p                Print "十" if the "ten" digit is non-zero, nothing otherwise
       ,A%sn@]z?       Get the character specified by the last digit if that digit is non-zero, empty string otherwise
                ' +q   Append a space and print

Alternative version (Stax 1.0.6), 59 bytes (by @recursive)

This uses a feature that is inspired by this challenge and is only included in Stax 1.0.6 which postdates the challenge.

éz░╖▐5à{│`9[mLùÜ•ëO╞îπl▼Γ─§╥|▒╛Δ◙Φµ'r╠eƒÿQ╫s♪Ω]£ï♪D3╚F◙δÿ%‼

The ASCII version is

9mX{x\_x*YA/]yA-y20<*!*+y9>A*+yA%]0-+"NT|,,t.%,p&()(!'^pq kzi !X6"!s@mJ

This version constructs the index array and then uses it to index the string of Chinese characters to avoid redundant stack operations (c,a,n) and multiple @s.

Explanation

9mX{...m    Loop `n` times and map `1..n` to a list of strings, `n`=1..9
        J   Join the strings with space and print with a newline

x\_x*YA/]yA-y20<*!*+y9>A*+yA%]0-+"..."!s@
x\                                           A pair: (inner loop variable, outer loop variable)
  _x*Y                                       Product of inner and outer loop variable
      A/                                     floor(product/10)
        ]                                    [floor(product/10)]
         yA-                                 Product does not equal 10
            y20<                             Product is less than 20
                *!                           `Nand` of them
                                             This is true (1) if the product is in the range {10}U[20,81]
                  *                          Repeat [floor(product/10)] this many times
                                             This results in itself if the predicate above is true, or empty array if it is false
                   +                         Add it to the list of [inner loop var, outer loop var]
                                             This list will be used to index the string "得一二三四五六七八九十"
                    y9>A*                    Evaluates to 10 if the product is larger than 9, 0 otherwise
                                             When indexed, they become "十" and "得", respectively
                         +                   Append to the list of indices
                          yA%                Product modulo 10
                             ]0-             [Product modulo 10] if that value is not zero, empty array otherwise
                                +            Append to the list of index
                                 "..."!      "得一二三四五六七八九十"
                                       s@    Index with constructed array
                           

Python 3, 151 149 146 bytes

-3 bytes thanks to Rod.

l=" 一二三四五六七八九"
for i in range(1,10):print([l[j//i]+l[i]+('得',l[j//10][10<j<20:]+'十')[j>9]+l[j%10]for j in range(i,i*i+1,i)])

Try it online!

Javascript, 190 bytes

(_="得一二三四五六七八九十")=>{for(i=1;i<10;i++){for(t="",v=0;v<i;t+=_[++v]+_[i]+[...(v*i+'')].map((a,b,c) => c.length>1&&b==0?(a>1?_[a]+'十':'十'):b==0?'得'+_[a]:_[a]).join``+' ');console.log(t)}}

a=(_=" 一二三四五六七八九")=>{for(i=1;i<10;i++){for(t="",v=0;v<i;t+=_[++v]+_[i]+[...(v*i+'')].map((a,b,c) => c.length>1&&b==0?(a>1||c[1]==0?_[a]+'十':'十'):b==0?'得'+_[a]:_[a]).join``+' ');console.log(t)}}
a()