Name for this pattern, representing one-to-many relationships by adding columns with an index to a single table

I don't know that there's an official or technical name for the relationship you're describing (I also don't know if you'll find many pet lovers who would like to be restricted to one dog, one cat, etc.), but here is how I would probably approach this:

CREATE TABLE dbo.People
(
  PersonID INT PRIMARY KEY
  --, ... name, etc.
);

CREATE TABLE dbo.PetTypes
(
  PetTypeID INT PRIMARY KEY
  --, ... species name, etc.
);

CREATE TABLE dbo.PeoplePets
(
  PersonID INT, -- FK to dbo.People
  PetTypeID INT, -- FK to dbo.PetTypes
  --, ... pet name etc.
  PRIMARY KEY (PersonID, PetTypeID)
);

Now whether this is simple enough, I'm not sure. You might have different attributes for different types of pets, and so you may need to extend this, but I think in general the concept is sound.

First off, apply normalization to the table and make sure that it really is one table with 170 attributes and not several tables jammed together.

Then decide whether your table will be sparsely populated and if so, decide whether storage space is a concern that warrants creating sub-tables to hold subsets of nullable columns.

If you do decide to create sub-tables, remember to factor in the extra index space when calculating whether a sub-table saves storage or not.