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I am confused by the use of meine in this quote from Die Leiden des jungen Werther
Waren nicht meine übrigen Verbindungen recht ausgesucht vom Schicksal, um ein Herz wie das meine zu ängstigen?
I take ein Herz wie das meine to mean a heart such as mine.
If meine refers to Herz shouldn't it be in the accusative neuter?
ein Herz wie das meinem
No. It's perfectly right. You are also right supposing that the case is accusative and the gender neuter. You are wrong, however, using meinem (is dative).
In the Wiktionary
table, cf. Nicht-attributiv, mit Artikel, you'll find agreement with Goethe's text. It's probably another register (gestelzt) but definitely not wrong (nor colloquial).
This use is as much antiquated as it is lyrical. It definitively is not colloquial. Some may use that occasionally but its by far not the norm.
In your example, meine is a possessive adjective with a regular weak ending -e (just as in das kranke Herz). The more modern form would be
ein Herz wie meins
with a neuter possessive pronoun (ending -s). Personally, I do find the combination of definite article and possessive adjective old-fashioned. Note that these adjectives can have -ig appended to them.
daß grade Athen der Ort war wo ein Geist wie der seinige sich aufs herrlichste entwickeln mußte (Quelle)
wie das meinemis neither accusative nor right, it is a mix with dative and wrong. I cannot find any "sounds odd/old/wrong" on your quote from Goethe. just the entire way of speaking in this book is a bit "gestelzt".