What could this mean - "you're turning into a suit on me".?

It can mean stuffy & serious
official/more mature

U re clingy

Its also another used by gangsters to describe a fed, like the FBI.

When you put on a suit you create a different look. It could have different effects. It could make you look beautiful and stylish, interesting, comfortable, a feeling of being dressed. It could also mean that you are becoming part of him like a suit you put on every day. They are positive things. The negative connotation could be , that you turning into a suit on him and it feels like cramping him or too much. Too restricted , too hot or too uncomfortBle. I suppose it depends on the tone it was said. If it was said with a smile or in an annoying tone.

you're getting really corporate. less human. more robotic

All over him/her like a suit.., give them some space