Why would a Shakespearean audience relate to Shylock differently than a modern audience would?
It's difficult to say precisely how an audience in Shakespeare's day would have reacted to something as opposed to a modern audience, because it will always be a gross generalisation. Think of one modern audience - within five hundred people, there will be a huge variety of various and opposing opinions on any given topic. To generalise about a whole generation of audiences is many times more absurd than ultimately generalising about any single one.
That said, it's clear that Shakespeare's audience weren't uncomfortable, or as uncomfortable with racism or racial stereotypes as our audience are. If you read Christopher Marlowe's play The Jew of Malta, you'll see that Marlowe's presentation of Barabas (another Jew, and a forerunner of Shylock) is almost grotesquely comic.
The persecution of the Jews only really became generally opposed after the events of the Holocaust and World War 2: before that, widespread hatred and persecution of the Jews stretched well back beyond the Elizabethan age (you can even find it in Chaucer's Prioress' Tale).
So it wouldn't be impossible to argue that a negative portrayal of Shylock would not offend an Elizabethan audience in general as much as it might offend a modern one. You might even think an Elizabethan audience would take pleasure in Shylock's demise (remember The Merchant of Venice has a - largely - happy last act). But of course, it all depends how you interpret Shakespeare's presentation of Shylock.
A Shakespearean audience would tend to be much more provincial in their attitude towards Shylock than a modern audience. Remember, in the Elizabethan Age, news traveled slowly and cultures were slower to change than today in our information, technology driven age. Deep rooted, stereotypical prejudices existed towards the Jews, so we have Shylock as the typical usurer (moneylender).
Unfortunately, we still have prejudice in today's world; however, many scholars and audiences would argue that Shylock is not all bad. In fact, a large question that looms over this play is whether or not Shylock truly deserved his punishment. Is Portia's trickery justified? We live in an age after World War II and the Holocaust, so that should, most definitely affect our judgement on Shylock's character and the historical persecution of the Jews.
In 16th century Europe usury, the lending of money for interest, was banned for Christians (it still is for Muslims), but not for Jews. Of course Christian commerce needed banking services and generally speaking the Jews provided it. So the Jews were often 1) rich and 2) foriegn. This caused tension, jealousy, mistrust, etc.
Also, European history is littered with outbreaks of religious extremism. 'The Jews are heretics who murdered Christ and should be converted or killed.' Was this a good excuse to steal their wealth and pick on an undefended foriegn community, possibly.
Christian-Jewish relations were generally poor. In fact, in Venice, the Doge allowed Jews to settle (because they needed bankers) but only in an area of old factories called La Ghetto, (the original 'ghetto'). They were not treated badly by the standards of the day and had some protection in law, but they were outsiders and, as today, people had many ignorant suspicions about their motives and practices.
Probably the average European Christian was anti-semetic. It's a tribute to Shakespeare that he shows the relationship between Jews and Christians from both sides of the line and has two wonderful monologues where Shylock shows us the world from his side as a persecuted, 'spat on', kicked around, minority with feelings, needs, loves, griefs.
In the context of the day, this was radical multi-culturalism.