Took me a minute, but you are definitely

A pair of shoes

I begin with the last.

Either a new pair of shoes begins where the last pair left off, or that each step begins with the previous.

I can make 1 + 1 = 1.

1 shoe + 1 shoe = 1 pair of shoes

If you miss me, then I will make you shy.

If you forget your shoes you will be shy

My other side can be used as identification.

footprints are used in investigations

I can make things smaller.

Stepping on things can make them smaller. Also the fact that they make you taller can make other things smaller.

Sometimes I can upset you.

Tripping on shoelaces can aggravate you.

Took me a minute, but you are definitely

A pair of shoes

I begin with the last.

A pair of shoes can be made with a last -- Credit to @h34

I can make 1 + 1 = 1.

1 shoe + 1 shoe = 1 pair of shoes

If you miss me, then I will make you shy.

If you forget your shoes you will be shy

My other side can be used as identification.

footprints are used in investigations

I can make things smaller.

Stepping on things can make them smaller. Also the fact that they make you taller can make other things smaller.

Sometimes I can upset you.

Tripping on shoelaces can aggravate you.