Some confusion over the definition of achievement tests and the objectives of such tests is evident in these conversations.
Achievement tests purport to identifty what a student has learned academically.
IQ tests have a more ambitious goal of attempting to estimate an individual's learning potential i.e. what they are capable of learning. This is true whether or not the individual is a native English speaker. A teacher can use scores on an achievement test to identify Independent Learning level, Instructional learning level and Frustration level. It is possible to conceive of a regular school achievement test booklet that could have a math test which is too simple for the student and reading and language tests which are too difficult. This would suggest that the best achievement tests for use with ESL students might be good quality, individually administered, wide range tests like the Woodcock-Johnson Achievement Test, Kaufman Test of Educational Achievement, Peabody Individual Achievement Test, Wechsler Individual Achievement Test, etc.