You deduced right, bash variables by default contain strings, and its values are treated as strings.
You can use the declare built-in command to explicitly say they store integers (declare -i myintvar), or indexed arrays (declare -a myarr), or associative arrays (declare -A mymap), etc., but not booleans.
The closest you can get to booleans is to use integer values 0 and 1 and evaluate expressions as arithmetic expressions with the (( expr )) command (bash-specific), or with arithmetic expansion $(( expr )) (POSIX-compatible). Those commands evaluate expr according to rules of shell arithmetic.
For example:
A=1
B=0
(( C = \!A )) # logical negation ==> C = 0
(( D = A && B )) # logical AND ==> D = 0
E=$(( A ^ B )) # bitwise XOR ==> E = 1
In bash, you can also use declare -i and let:
declare -i E='A||B' # equivalent to: E=$((A||B)), or ((E=A||B))
let C='!A' # equivalent to: C=$((\!A)), or ((C=\!A))
which are a longer way of saying ((..)) or $((..)). They both force arithmetic evaluation of the expressions given.
Note that ! has a special meaning in most shells (including bash), it causes history expansion. To prevent it, we must escape it with a backslash, or quote it.