I see the TM tag with smaller text than it's associated name, does the same apply for the ® symbol.
Should it be the same font size as the name or smaller like exponents are displayed in math?
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The ® symbol doesn't need to be in a smaller font size than the actual text. In Name™ the trademark symbol ™ is not a smaller font size but a special character which has a shape that is smaller than the characters T and M.
Most of the good fonts have a wide range of unicode support which means they have glyphs/letters drawn specifically for these purposes. In the following image you can see how some fonts like Source Sans Pro and Mukta have a small RTM symbol by default.
So you can use it as a superscript if you feel it's attracting too much attention (e.g. Logo design), but when using in a running text it's better to use it the way it's designed.
The USPTO shows examples of the symbol used in copy (the text on a page) that are the same size and typeface as the regular text.
Additionally, the usage and examples in the Trademark Manual of Examining Procedure (TMEP) would also lead one to infer that the symbol is used in the same size and typeface as the accompanying text:
So, the clear answer to "Does the ® Symbol need to be in a smaller font size than the actual text?" is, "No".
However, it is not explicitly stated in the official guides and instructions on use that it must be the same size, so this also seems to indicate that is may be smaller if you'd like it to be.
Keep in mind that there is also an expectation that, when used, "the notice should be in a manner and location that is conspicuous and won't be missed by a casual observer". This seems to be lost on many logo designers. Even the examples used by the USPTO show the registered symbol next to some graphics to be not much bigger than a periods used in the text on the same page.