There is no function that can locate a name in the middle of an array of names. All of the lookup type functions look in a one-dimensional range only (a single column or a single row). You have two choices that I can think of.
First idea:
If you have only a few managers, you can search each column separately to find a match. You only need to know that there is a match, not what row it is on so you can use COUNTIF instead of a lookup. But if it gets to be more than a few, the formula gets long.
Formula in Table 2 column B =IF(COUNTIF(Table 1::A,A)>0,Table 1::A$1,"")&IF(COUNTIF(Table 1::B,A)>0,Table 1::B$1,"")&IF(COUNTIF(Table 1::C,A)>0,Table 1::C$1,"")
Second idea:
Using formulas, transform Table 1 so it has a column for employee names and a column for their manager names. Then you can use XLOOKUP to get your results from this table. Here is one way to do the transformation.

A2 =MOD(ROW()−2,ROWS(Table 1::A)−1)+2
B2 =QUOTIENT(ROW()−2,ROWS(Table 1::A)−1)+1
C2 =INDEX(Table 1::A:C,A2,B2)
D2 =INDEX(Table 1::$1:$1,B2)
fill down with all to complete the table.
If you add new columns to Table 1, do it in the middle not to the far right or left so that the new columns will be automatically included in the range used in the formula in column C
Make sure Table 1-1 is long enough. You may want to include a lot of extra rows for future expansion. Extra rows won't hurt anything, you'll just get error triangles.
For Table 2, the lookup formula is
=XLOOKUP(A2,Table 1-1::C,Table 1-1::D,"",0)
You can cut Table 1-1 and paste it on a different sheet to get it out of sight.