cv.glmnet returns an object of type cv.glmnet (using dots in class names is asking for trouble with method dispatching, but this does not seem to be the problem here), which has overwritten methods for predict and coef, namely predict.cv.glmnet and coef.cv.glmnet. When you read their docs with

> ?coef.cv.glmnet
> ?predict.cv.glmnet

you will notice that they have an option s for specifying the lambda used. The coefficients seen in your question refer to s="lambda.1se". Set it to s="lambda.min", and the result is identical with your model trained with lambda.min:

> coef(cv.ridge, s="lambda.min")
19 x 1 sparse Matrix of class "dgCMatrix"
                        1
(Intercept) -1.468326e+03
(Intercept)  .           
PrivateYes  -5.278781e+02
Accept       1.004588e+00
...
> pred.out <- predict(cv.ridge, model.matrix(Apps~.,College), s="lambda.min")
> mean((pred.out- College$Apps)^2)
[1] 1358455

Remark: What do your data sets train and test contain? Apparently somethind different from College because your MSE and coefficients are different.

cv.glmnet returns an object of type cv.glmnet (using dots in class names is asking for trouble with method dispatching, but this does not seem to be the problem here), which has overwritten methods for predict and coef, namely predict.cv.glmnet and coef.cv.glmnet. When you read their docs with

> ?coef.cv.glmnet
> ?predict.cv.glmnet

you will notice that they have an option s for specifying the lambda used. The coefficients seen in your question refer to s="lambda.1se". Set it to s="lambda.min", and the result is identical with your model trained with lambda.min:

> cv.ridge <- cv.glmnet(model.matrix(Apps~.,College), College$Apps, alpha=0, nfold=100)
> coef(cv.ridge, s="lambda.min")
19 x 1 sparse Matrix of class "dgCMatrix"
                        1
(Intercept) -1.468326e+03
(Intercept)  .           
PrivateYes  -5.278781e+02
Accept       1.004588e+00
...
> pred.out <- predict(cv.ridge, model.matrix(Apps~.,College), s="lambda.min")
> mean((pred.out- College$Apps)^2)
[1] 1358455

Remark: What do your data sets train and test contain? Apparently something different from College because your MSE and coefficients are different.