Is LR62 the same as LR516?

Yes. LR62 is another code for the same cell: alkaline, 1.5 V, 5.8 × 1.6 mm.

Can I use SR516SW instead of LR516?

Upgrade: steadier voltage, longer life.

Same 5.8 mm diameter, but 2.1 mm tall instead of 1.6 mm. Only if the holder fits; thinner cells can lose contact.

Same 5.8 mm diameter, but 2.7 mm tall instead of 1.6 mm. Only if the holder fits; thinner cells can lose contact.

LR516 is a 1.5 V alkaline cell with a typical capacity of 11 mAh.

5.8 mm across and 1.6 mm thick.

LR516 is the same cell as LR62, so any of those codes will fit; SR516SW is the same size in silver oxide and holds a steadier voltage for longer.

IEC designation LR516.

LR516 at actual relative size: 5.8 mm across, 1.6 mm thick. The dashed circle is a US quarter, 24.26 mm across (US Mint coin specifications).

The same cell under another code. Any of these fits, whatever the brand.

Same size, different chemistry. Read the verdict before you buy.

Cell	Chemistry	Voltage	Verdict
SR516SW	silver oxide	1.55 V	Upgrade Upgrade: steadier voltage, longer life.

Same diameter, different thickness. These are not drop-in replacements.

Cell	Size	Verdict
LR521	5.8 × 2.1 mm	Same 5.8 mm diameter, but 2.1 mm tall instead of 1.6 mm. Only if the holder fits; thinner cells can lose contact.
LR527	5.8 × 2.7 mm	Same 5.8 mm diameter, but 2.7 mm tall instead of 1.6 mm. Only if the holder fits; thinner cells can lose contact.

Alkaline version of the same size; voltage sags as it drains.

Is LR62 the same as LR516?: Yes. LR62 is another code for the same cell: alkaline, 1.5 V, 5.8 × 1.6 mm.
Can I use SR516SW instead of LR516?: Upgrade: steadier voltage, longer life.
Will LR521 work in place of LR516?: Same 5.8 mm diameter, but 2.1 mm tall instead of 1.6 mm. Only if the holder fits; thinner cells can lose contact.
Will LR527 work in place of LR516?: Same 5.8 mm diameter, but 2.7 mm tall instead of 1.6 mm. Only if the holder fits; thinner cells can lose contact.
What voltage is LR516?: LR516 is a 1.5 V alkaline cell with a typical capacity of 11 mAh.
What size is LR516?: 5.8 mm across and 1.6 mm thick.