Comparing two groups of individuals I am trying to show that 2 groups of individuals are intrinsically the same. I have various characteristics to measure for both group A and group B, such as height, weight, age, etc. I have taken the averages of both groups, and by inference I can see that they are similar - is there a more robust way of doing this?
It sounds like you've basically eyeballed the groups to decide they're similar.  A more advanced way of eyeballing would be to plot the distribution of each characteristic to see how similar they look.

In order to move beyond eyeballing, though, you'd need recourse to statistics.  One common method--that comes up frequently in two-group studies--is to do a statistical test to verify that the groups are similar.

Broadly speaking, you'd look at each characteristic, say height, and assert your hypothesis: average height is the same in both groups.  Then you could do a hypothesis test to see if the average height is actually the same or if it's different.  The specific statistical test you'd use depends on the data in your possession.

For example, see the t-test or the chi-squared test for a difference in means (or just search for "difference in means test").

You'd run the test (statistical software generally has a feature like this) to determine if it shows a significant difference.  If so, you might be comfortable saying the groups are not similar inasmuch as that characteristic (height) is concerned.

Then you'd repeat the procedure for weight and each of the other characteristics.

At the end, many researchers make a little table that shows the $p$-values (which you get from running the test) for each of the characteristics.  This gives your readers the ability to see and decide for themselves whether the groups are similar.