Uniqueness of hyperreals contructed via ultrapowers

The construction I've seen of the field of hyperreal numbers considers a non-principal ultrafilter $\mathcal{U}$ on $\mathbb{N}$, then takes the quotient of $\mathbb{R}^{\mathbb{N}}$ by equivalence respect to $\mathcal{U}$, that is, $(a_{n})=(b_{n})$ if the set $S$ of indices $n$ for which $a_{n}=b_{n}$ is inside $\mathcal{U}$.

Is it clear (or even true) that one obtains an isomorphic field with a different choice of ultrafilter?

No, one has to invoke rather strong hypotheses such as the Continuum Hypothesis (CH) to obtain uniqueness. The classic paper on this is the following, where it is proved that it is consistent with ZFC that there are $\,2^{\large \aleph_0}$ non-isomorphic hyperreal fields.

Judy Roitman. Non-Isomorphic Hyper-Real Fields from Non-Isomorphic Ultrapowers.
Math. Z. 181, 93-96 (1982)

See also N. Aldenhoven's 2010 Bachelor's thesis Uniqueness of the Hyperreal Field where you can find an elementary exposition of an earlier result of Erdos, Gillman and Hendrikson (1955) that all real-closed fields of the same cardinality as $\,\Bbb R\,$ with a $\eta_1\!$-ordering are isomorphic.

More recent results can be located by searching for citations of Roitman's paper.

I would like to add to Bill's answer:

Judy Roitman found a particular model of ZFC$+ \neg$CH where there are $\mathfrak c$ many different hyperreal fields. Very recently, it was shown that in every model of ZFC$+ \neg$CH there actually exist $2^\mathfrak c$ different hyperreal fields... as many as you could hope for since there are exactly $2^\mathfrak c$ (free) ultrafilters on $\omega$!

A dichotomy for the number of ultrapowers

Ilijas Farah & Saharon Shelah

Journal of Mathematical Logic 10 (01n02):45-81 (2010)