Compare two files and count number of matching lines

Dear All,
I would like to compare two files and return the number of matches found.
Example

File A

Lx2
L1_Mus1
L1Md_T
Lx5
L1M2
L1_Mus3
Lx3_Mus
Lx9
Lx2A
L1Md_A
L1Md_F2

File B

L1_Mus3
L1_Mus3
L1_Mus3
L1_Mus1
L1_Mus1
L1_Mus1
L1_Mus1
L1_Mus1
L1_Mus1
L1_Mus1
L1_Mus1
L1_Mus1
L1_Mus1
L1_Mus1
L1_Mus1
L1_Mus1

Desidered Output
File A

Lx2   0
L1_Mus1   13
L1Md_T    0
Lx5   0
L1M2   0
L1_Mus3   3
Lx9   0
Lx2A   0
L1Md_A   0
L1Md_F2   0

Could You please help me?

Many thanks,
Paolo

try

awk 'NR==FNR{a[$0]++;next}{if ($0 in a){print $0,a[$0]} else {print $0,0}}' file B file A

$ awk 'FNR==NR{ A[$1]++; next}{ print $1 , A[$1]+0 }' fileB fileA

Many many thanks,
Paolo