You can prefix the table names with the database name to identify the two similarly named tables. You can then use that fully qualified table name to refer to the similarly named fields.
So, without aliases:
select db1.table1.id, db1.table1.value1, db2.table1.value1
from db1.table1 inner join db2.table1 on db1.table1.id = db2.table1.id
and with aliases
select t1.id, t1.value1, t2.value1
from db1.table1 as t1 inner join db2.table1 as t2 on t1.id = t2.id
You may also want to alias the selected columns so your select line becomes:
select t1.id as id, t1.value1 as value_from_db1, t2.value1 as value_from_db2